ON SOME LINEAR CODES OVER Z2S A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by MANISH K. GUPTA to the DEPARTMENT OF MATHEMATICS INDIAN INSTITUTE OF TECHNOLOGY, KANPUR December, 1999 ✁ ✂ ✄ ☎✆✝✞✟✠ ✡ ☛ ☞✝✌✠ ✍✞✝ ✆✟ ✎ ✏✑✡✠ ✞✝ iv CERTIFICATE It is certified that the work contained in the thesis entitled On Some Linear Codes over Z2s by Manish K. Gupta has been carried out under our supervision and that this work has not been submitted elsewhere for a degree. Professor M. C. Bhandari Professor A. K. Lal Department of Mathematics Indian Institute of Technology December, 1999 Kanpur v vi Nomenclature S : A Commutative ring with unity |S| : Cardinality of the set S Sn : Set of all n-tuples over S Fq : Set of q Alphabets Fqn : Set of all n-tuples over Fq M : A left S-module GF (q) : Galois field with q elements Zq : Ring of integers modulo m Zps : Ring of integers modulo ps U : Set of all units of Zps Z : Set of all zero divisors of Zps ⌊x⌋ : Greatest integer less than or equal to x ⌈x⌉ : Smallest integer greater than or equal to x x : Element of Fnq wH (x) : Hamming weight of x wL (x) : Lee weight of x wE (x) : Euclidean weight of x dH (x, y) : Hamming distance between x and y dL (x, y) : Lee distance between x and y dGL (x, y) : Generalized Lee distance between x and y dH : Minimum Hamming distance of C dL : Minimum Lee distance of C dGL : Minimum generalized Lee distance of C vii d(x, D) : Smallest distance of x from a word in D C : A code of length n c : A codeword of C (n, M, dH , dL , dE ) : A code of length n, number of codewords M and minimum Hamming, Lee and Euclidean distances respectively dH , dL and dE [n, k, dH , dL, dE ] : A Linear code of length n, 2-dimension k and minimum distances dH , dL and dE respectively [n, k, dH , dL, dGL ] : A Linear code of length n, 2-dimension k and minimum distances dH , dL and dGL respectively [n, k] : A Linear Code AH (i)(AL (i)) : Number of codewords of Hamming ( Lee ) weight i dr (C) : r th Generalized Hamming weight (GHW) of C wS (D) : Support size of C C⊥ : Dual code of C Aut(C) : Automorphism group of C HamC (x, y) or WC (x, y) : Hamming weight enumerator (hwe) of C ωj (c) : |{k : ck = j}| cweC (x0 , . . . , xm ) : Complete weight enumerator (cwe) of C sweC (x, y, z) : Symmetrized weight enumerator (swe) of C LeeC (x, y) : Lee weight enumerator of C Sk (q) : q-ary Simplex Code Gk (q) : Generator matrix of Sk (q) Mk,u (q) : q-ary MacDonald Code Gk,u (q) : Generator matrix of Mk,u(q) Sk : Binary Simplex Code Sˆk : Extended binary Simplex Code G(Sk ) : Generator matrix of Sk viii G(Sˆk ) : Generator matrix of Sˆk RM(r, m) : r th -order Reed Müller Code P (m) : Binary nonlinear Preparata Code K(m) : Binary nonlinear Kerdock Code A(n, dH ) : max{M| there exists a binary (n, M, dH ) code } φ : Gray map φ(C) : {φ(c) : c ∈ C} : Gray image of C h2 (x) : Irreducible polynomial over Z2 such that it divides xn − 1 over Z2 h(x) : Hensel uplift of h2 (x) over Z4 B : p-basis of C p−dim(C) : p-dimension of C φG : Generalized gray map φG (C) : Generalized gray image of C Cφ : Generalized gray image of C in 2-basis form C (1) : Binary residue code of C C (2) : Binary torsion code of C θ(c) : Correlation of c ∈ C Skα : Simplex Code of type α Skβ : Simplex Code of type β Gk : Generator matrix of Skα Gβk : Generator matrix of Skβ S¯kα : Punctured code of Skα obtained by deleting the zero coordinate Ri : Rows of a generator matrix K4m : Klemm code O8 : Octacode code QRn : Uplifted extended quadratic residue code over Z4 R1,m−s+1 : First order Reed Müller code over Z2s Cp : Norm quadratic residue (NQR) code ix Hp : Finite version of Poincáre upper half-plane δ : Arbitrary chosen quadratic non residue modulo p γ : First row of a circulant matrix C Pγ (z) : Polynomial corresponding to γ P SL2 (p) : Projective special linear group on GF(p) SL2 (p) : Special linear group on GF(p) R(C) : Covering radius of C N (i) : Norm of C with respect to ith -coordinate N(C) : Norm of C : End of the proof Synopsis Name of the student: Manish Kumar Gupta Degree for which submitted: Ph.D. Roll No.: 9310863 Department: Mathematics Thesis Title: On Some Linear Codes over Z2s Name of the Thesis Supervisors: Prof. M. C. Bhandari & Prof. A. K. Lal Month and Year of Thesis Submission: December, 1999 Let q be a positive integer and let Fq be a set of q-alphabets. A code C of length n and size M is a subset of Fnq having M elements. The elements of C are called codewords. In order to be able to correct errors we associate some algebraic structure with Fq . If q is a prime power one usually take Fq = GF (q) otherwise Fq = Zq . Let Fq = GF (q)( Zq ). A linear code of length n over GF (q)( Zq ) is a subspace (submodule) of Fnq . Two codes over Zps are said to be equivalent if one can be obtained from the other by a permutation of coordinates and or by multiplying one or more coordinates by a unit in Zps . Let C be a linear code of length n over Zps . In 1995, Calderbank and Sloane have shown that C is equivalent to a code generated by the rows of the matrix (see[15])      G=    I k0 A01 A02 ··· A0s−1 A0s 0 pIk1 pA12 ··· pA1s−1 pA1s 0 .. . 0 .. . p 2 I k2 .. . ··· p2 A2s−1 .. . p2 A2s .. . 0 0 0 ··· ps−1 Iks−1 ps−1 As−1s .. .      ,    (1) xi where Aij are matrices over Zps and the columns are grouped into blocks of size k0 , k1 , · · · , ks−1 , ks . Let k = Ps−1 i=0 (s − i)ki . Then |C| = pk . Note that C is a free module if and only if ki = 0 for all i = 1, 2, . . . , s − 1. Notion of dual code of C is defined via inner product x · y = Pn i=1 xi yi (mod ps ). Two notorious binary nonlinear codes–the Kerdock code and the Preparata code-have in the past been something of an enigma in coding theory since they possess many linear-like properties. Until recently, their mysterious formal duality (with respect to the MacWilliams transform) had been dismissed as a “mere coincidence” since an appropriate algebraic notion of duality was lacking for nonlinear codes. In[52], Hammons et al have shown that both Kerdock and Preparata codes are gray images of some linear codes over Z4 which are dual to each other over Z4 . Description of these codes become simpler using the properties of corresponding linear codes over Z4 . In addition, the customary notions of generator matrices, parity-check matrices, syndromes etc., become meaningful. Since then many linear codes over Z4 and in general over Zps have been investigated by several researchers. They have also studied the Gray images of different codes. Let C be a linear code over Z4 . Then C has a generator matrix of the form   G= Ik0 A 0 2Ik1  B1 + 2B2  2C , (2) where A, B1 , B2 , C have entries 0 and 1 and Ik is the identity matrix of order k. Let dH (dL ) be the minimum Hamming (Lee) distance of C. In[93], E. M. Rains has shown that dH ≥ ⌈ d2L ⌉. C is said to be of type α ( β) if dH = ⌈ d2L ⌉ (dH > ⌈ d2L ⌉). The Gray map φ : Z4n → Z22n is the coordinate wise extension of the function from Z4 to Z22 defined by 0 → (0, 0), 1 → (0, 1), 2 → (1, 1) and 3 → (1, 0). The image φ(C) = {φ(c) : c ∈ C} of C under the Gray map φ is a binary code of length 2n. xii However it may not be linear. If φ(C) is linear then C is called Z2 -linear. A binary code of even length is called Z4 -linear if it is equivalent to φ(C) for some linear code over Z4 . In[53], Hammons et al have shown that many known binary nonlinear codes are Z4 -linear. In the present dissertation we have constructed Simplex codes of type α and β over Z4 and over Z2s , and first order Reed Müller code over Z2s . Some fundamental properties like 2-dimension, weight hierarchy etc. are obtained. We have also considered their gray images in two ways and obtained some interesting binary codes. It is shown that Simplex code of type α and β and many known self dual codes satisfy the chain condition. A brief survey of known results is given in Chapter 2. A vector v is a p-linear combination of the vectors v1 , v2 , . . . , vk if v = l1 v1 +. . .+lk vk with li ∈ Zp for 1 ≤ i ≤ k. A subset S = {v1 , v2 , ..., vk } of C is called a p-basis for C if for each i = 1, 2, ..., k − 1, pvi is a p−linear combination of vi+1 , ..., vk , pvk = 0, C is the p-linear span of S and S is p-linearly independent[112]. The number of elements in a p-basis for C is called the p-dimension of C. It is easy to verify that the rows of the matrix B=        Ik0 A 2Ik0 2A 0 2Ik1  B1 + 2B2  2B1 2C      (3) form a 2-basis for the code C generated by the matrix G given in (1). A linear code C over Z2s ( Z2 ) of length n, 2-dimension k, minimum Hamming distance dH and minimum Lee distance dL is called an [n, k, dH , dL ] ([n, k, dH ]) or simply an [n, k] code. Let φ(B) be the matrix obtained from B (given in (3)) by applying the gray map φ to each row of B. The code Cφ generated by φ(B) is a [2n, k, ≥ ⌈ d2L ⌉] binary linear code. xiii Let C be an [n, k] linear code over Z2s . For 1 ≤ r ≤ k, the r-th generalized Hamming weight of C is defined by dr (C) = min{wS (Dr ) : Dr is an [n, r] subcode of C}, where wS (D), called support size of D, is the number of coordinates in which some codeword of D has a nonzero entry. The set {d1 (C), d2 (C), . . . , dk (C)} is called the weight hierarchy of C. C is said to satisfy the chain condition if there exists a chain D1 ⊆ D2 ⊆ · · · ⊆ D k , of subcodes of C satisfying wS (Dr ) = dr (C), 1 ≤ r ≤ k. Let Gk and Gβk be defined inductively by   G1 = [0123], Gk =     Gβ2 =    1111 0 2  0123 1 1 ,   00 · · · 0 11 · · · 1 22 · · · 2 33 · · · 3   Gk−1 Gk−1   Gβk =   Gk−1 Gk−1 ,  11 · · · 1 00 · · · 0 22 · · · 2  Gk−1 Gβk−1 Gβk−1 .  Note that columns of Gk consist of all elements of Zk4 and no two columns of Gβk are multiples of each other. The code Skα (Skβ ) generated by Gk (Gβk ) over Z4 is of type α (β) and is called Simplex code of type α (β). In[97], Satyanarayana has shown that the Lee weight of every nonzero codeword of Skα is 22k and in[20] Carlet has classified all constant Lee weight codes over Z4 . For each a ∈ Z4 , let ā be the reduction of a modulo 2 then the code C (1) = {(c¯1 , c¯2 , . . . , c¯n ) : (c1 , c2 , . . . , cn ) ∈ C} is a binary linear code called the residue code   of C. It is shown that the residue code of Skα Skβ is equivalent to 2k copies of the extended binary simplex code ( 2k−1 copies of the binary simplex code). The following theorems that give properties of Skα and Skβ are proved in Chapter 3. Theorem 0.1 The Hamming and Lee weight distribution of Skα are: xiv 1. AH (0) = 1, AH (22k−1 ) = 2k − 1, AH (3 · 22k−2 ) = 2k (2k − 1) and 2. AL (0) = 1, AL (22k ) = 22k − 1. where AH (i)(AL (i)) denotes the number of codewords of Hamming (Lee) weight i in Skα . Theorem 0.2 The Hamming and Lee weight distributions of Skβ are: 1. AH (0) = 1, AH (22k−2 ) = 2k − 1, AH (2k−3 [3(2k − 1) + 1]) = 2k (2k − 1) and 2. AL (0) = 1, AL (22k−1 ) = 2k − 1, AL (2k−1(2k − 1)) = 2k (2k − 1). Thus Skα is a [22k , 2k, 22k−1, 22k ] and Skβ is a [2k−1 (2k − 1), 2k, 22k−2, 2k−1(2k − 1)] linear code over Z4 . It is further shown that both Skα and Skβ are not Z2 -linear. Following theorems have been obtained for Skα and Skβ . Theorem 0.3 Skα satisfies the chain condition and its weight hierarchy is given by dr (Skα ) = r X i=1 22k−i = 22k − 22k−r , 1 ≤ r ≤ 2k. Theorem 0.4 Skβ satisfies the chain condition and its weight hierarchy is given by r dr (Skβ ) = n(k) − 2k−r−1(2k − 2⌈ 2 ⌉ ), 1 ≤ r ≤ 2k, where n(k) = 2k−1(2k − 1). The description of the code Cφ for the simplex codes of type α and β are given by the following theorems. Theorem 0.5 Let C = S¯kα , be the punctured code of Skα . Then Cφ is an [22k+1 − 2, 2k, 22k ] binary linear code consisting of two copies of binary simplex code S2k with Hamming weight distribution same as the Lee weight distribution of S¯kα . xv Theorem 0.6 Let C = Skβ . Then Cφ is the binary MacDonald code M2k,k : [22k − 2k , 2k, 22k−1 − 2k−1 ] with Hamming weight distribution same as the Lee weight distribution of Skβ . The gray image φ(C) of a linear code C over Z4 is a binary code of even length and the minimum Hamming distance as the minimum Lee distance of C. A natural Question arises : Does there exist a generalized gray map which maps a linear code over Zps to a code over Zp having similar properties? Recently Ana Sălăgean − Mandache has shown that it is not possible to construct a weight function on Zps for which Zps is isometric to Zps with the Hamming metric unless p = s = 2[98]. s−1 In[19], Carlet has introduced the generalized gray map φG from Z2s to Z2 2 and has obtained the Z2s version of Kerdock code. φG is a mapping from Z2s onto the Reed Müller code of order 1 and length 2s−1 . φG is defined as a Boolean function φG (u) evaluated on GF (2s−1) by : φG (u) : (y1 , y2 , · · · , ys−1) 7→ us + s−1 X ui yi . i=1 Note that φG is distance preserving[19]. We have used φG to obtain generalization of theorems 0.5 to 0.6 for codes over Z2s in Chapter 4. The concept of chain condition is extended to the codes over Zps in Chapter 2. In Chapter 5, we have shown that various known self dual codes over Z4 satisfy the chain condition. The construction of the first order Reed Müller code R1,m−s+1 of length 2m−s+1 is also given in Chapter 5. The following theorems are proved in Chapter 5. Theorem 0.7 D2m , E7 , E7+ and E8 satisfy the chain conditions. ′ Theorem 0.8 Klemm Code K4m (m ≥ 1) and the code K8 satisfy the chain condition and the weight hierarchy of K4m is given by dr (K4m ) =      r + 1, 1 ≤ r ≤ 4m − 2 4m, r = 4m − 1 & 4m. xvi Theorem 0.9 Extended quadratic residue codes QR8 and QR24 over Z4 satisfy the chain condition. Theorem 0.10 R1,m−s+1 satisfies the chain condition and its weight hierarchy is given by dt (R1,m−s+1 ) =  Pt−1 m−s−i   2 , 1≤t≤m−s+1 i=0   2m−s+1 , m − s + 1 < t ≤ m + 1. Theorem 0.11 R1,m−s+1 is Z2 -linear. Let p be an odd prime. The Poincaré finite upper half plane is the set ( Hp := x + √ δy x, y ∈ GF (p), y 6= 0, δ is a quadratic non residue (q.n.r.) ) (mod p) . In[110], Tiu and Wallace have used it to construct a new class of binary linear code Cp called norm quadratic residue (NQR) codes. The length of each codeword in Cp is p(p − 1), the number of points in Hp . The points in Hp are ordered lexicographically; i.e, (x, y) < (u, v) if x < u or x = u and y < v. For each a ∈ GF (p), the set Ba = {(a, 1), (a, 2), . . . , (a, p − 1)} is called a block. Let G be the (p + 1) × p(p − 1) matrix  G= p(p−1) where each Ri ∈ IF 2            R0  R1 .. . Rp is defined as follows.     ,     R0 has an 1 in position (x, y) if x2 − δy 2 , the norm of (x, y) in Hp , is a quadratic residue (q.r.) mod p and 0, otherwise. R1 , R2 , . . . , Rp−1 are block wise cyclic shifts of R0 corresponding to blocks B0 , B1 , . . . , Bp−1 and Rp is the all one vector. The code Cp generated by the matrix G is called the NQR Code[110]. If p is of the form 4m + 1 then Tiu and Wallace[110] have shown that dim Cp ≤ p, dH ≥ p − 1, Cp is xvii weakly self dual and that the weight of each codeword is divisible by 4. They conjectured that dim Cp = p. In Chapter 6, we show that if p is a prime of the form 4m − 1 then also Cp satisfies similar properties. The following results are proved for Cp . Theorem 0.12 dim Cp = p, (p 6= 2). Theorem 0.13 PSL2 (p) fixes Cp and acts transitively on the coordinate positions. Theorem 0.14 Cp is Z4 -linear. The Covering Radius R(C) of a linear code C is the least integer R such that spheres of radius R around the codewords cover the whole space. It is known that R(C) is the maximum weight of the coset leader. The following theorem gives the covering radius of Cp . Theorem 0.15 Covering radius of Cp is p(p−1) . 2 Moreover Cp is a normal code with every coordinate acceptable. The last chapter concludes with some general remarks. Acknowledgments It is better to do the right problem the wrong way than to do the wrong problem the right way. . . . Richard W. Hamming (1915-1998) The best way to learn a craft is to work under the guidance of a skilled professional. I am very grateful to my “guruji” Prof. M. C. Bhandari who has not only made me aware about the beautiful subject of “Algebraic Coding Theory” but also constantly supported me with his many ideas and healthy discussions. It has been a privilege to work with him. Learning how to do research and how to write technical papers from him has been a life time experiences in itself. I am also grateful to Prof. A. K. Lal for having accepted to be my co-guide. His critical comments and constructive suggestions have helped me to substantially improve the presentation of this thesis. I am highly indebted to both of my gurus for the patience they have shown in guiding and going through my work. This is also a unique occasion to express my sincere thanks to many people who have shared time to cooperate with and to help me. First of all I wish to thank Prof. R.K.S. Rathore for continuously inspiring me throughout my stay at IIT K. I also thank with pleasure Prof. U.B. Tewari, Prof. Shobha Madan, Prof. Prabha Sharma, Prof. A. K. Maloo and to all other members of the Mathematics Department, IIT K. I also thank to Prof. H.L. Janwa and Brajesh Kumar for helping me during my xviii xix visits at MRI, Allahabad. I am also thankful to C. Carlet, C. Charnes, C. Ding, M. Harada, Z. Quian, E.M. Rains, J. Wolfmann and J.A. Wood for sending the re/pre prints of their manuscripts. I wish to thank all my friends who has contributed to make my stay at IIT K as one of the most beautiful periods in my life. Let me express my sincere thanks to my colleagues and batch mates: Chanduka, Durairajan, K. Balaji, S.B. Rao, Mahesh, V.S.N. Kaliprasad and Nemade. It would be really immodest if i do not appreciate the various facilities provided by IIT Kanpur. Last but not least, it were the blessings of my parents and family members who have always shadowed me. Its really beyond words to express my feelings regarding them. Manish K. Gupta Contents Nomenclature vi Synopsis x Acknowledgments xviii 1 Introduction 1 2 Preliminaries and Survey 5 3 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Classical Codes over GF (q) . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Codes over Integer Residue Rings . . . . . . . . . . . . . . . . . . . . . 12 2.4 Codes over Zps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4.1 p-dimension of Linear Codes over Zps . . . . . . . . . . . . . . 16 2.4.2 Generalized Gray Map . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.3 Generalized Hamming weights of Linear codes over Zps . . . . . 21 2.4.4 Codes over Z4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Z4 -Simplex Codes and their Gray Images 25 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 26 Z4 -Simplex Codes of Type α and β . . . . . . . . . . . . . . . . . . . xx xxi 4 3.3 Gray Image Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Z2s -Simplex Codes and their Generalized Gray Images 36 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Z2s -Simplex Codes of Type α and β . . . . . . . . . . . . . . . . . . . 37 4.2.1 42 4.2 Gray Image Families . . . . . . . . . . . . . . . . . . . . . . . . 5 Codes Satisfying the Chain Condition 47 5.1 Self-Dual and Self Orthogonal Codes over Z4 . . . . . . . . . . . . . . 47 5.2 First order Reed Müller Code over Z2s . . . . . . . . . . . . . . . . . . 54 6 Norm Quadratic Residue Codes 6.1 Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusions Bibliography 57 57 65 66 Chapter 1 Introduction Men are from Mars, women are from Venus and computers are from hell thats what we say once they makes an error . . . , IEEE Computer Magazine Information Theory is a field in which profound theoretical investigation often has immediate and far-reaching impact on Technology and practice. Last year it was an exciting time for Information Theory distinguished to celebrate 50th year anniversary of Shannon’s seminal paper A Mathematical Theory of Communication[101]. Soon after Shannon’s paper Richard W. Hamming in 1950 published the paper[51] which together with Golay’s paper[41] forms the foundation of the constructive counterpart to Shannon’s theory, namely, Algebraic Coding Theory. Applications of Coding Theory ranges from deep space communications to consumer electronics. Infact, “ Algebraic Coding Theory is a branch of Engineering with roots in Mathematics and applications to Computer Science.” The origins of the Binary Code can be traced as far back as the 17th century, when the secretary of nature and the great philosopher of the world Sir Francis Bacon devised a binary scheme called Biliterarie Cipher or omnia per omnia a five letter binary code first mentioned in The Advancement of Learning (1605) and later published in detail in 1 2 De Augmentis Scientarum (1623) for encoding his diplomatic secret messages (see web site[4]). After this three men approached the subject from different directions. Early in the 19th century in France Joseph Marie Jacquard designed the first binary coded punched cards for operating Looms. The other two men were George Boole the English Mathematician whose algebra of propositional calculus forms the basis of the modern design of computer logic and Emile Baudot a french engineer whose cyclic-permuted code (now often called Gray code as it was patented by Frank Gray on 17.03.1953[46] ) represented a major advance in telegraphy. There are various other applications of this code like solving puzzles such as Tower of Hanoi and the Brain. It is very recent when it was used in solving a 30 year old coding theory puzzle[53]. The key idea of this puzzle can be described in few sentences. For many years it was observed that binary nonlinear Preparata and Kerdock codes satisfy the duality relations and they appeared to be duals to each other. Many researchers viz MacWilliams, Sloane etc. worked hard to explain this mystery. The answer turns out to be astonishingly simple. In 1994, Hammons et al have shown that both Kerdock and Preparata codes are gray images of linear codes over Z4 , which are dual to each other. This has motivated a great deal of research in Codes over Z4 and in general Codes over Zps [113], [15]etc. Though there were few papers on codes over the ring of integers modulo q in early seventies, it was forgotten by the researchers partly because of the presence of the zero divisors. These codes also found various other applications like orthogonal frequency division multiplexing (OFDM), phase modulated channels etc. to diverse areas such as bent functions and various lattice constructions. There are various binary linear codes studied so far by several researchers[79]. Some important class of binary codes are Hamming code, the first order Reed Müller code and Simplex code. Any nonzero codeword of the simplex code has many of the properties that we would expect from a sequence obtained by tossing a fair coin 2m − 1 times. 3 This randomness makes these codewords very useful in a number of applications such as range-finding, synchronizing, modulation, scrambling etc. Similar properties holds for the binary first order Reed Müller code. This family of code is one of the few for which maximum likelihood decoding is practical. Hamming code is the dual of the simplex code. All these codes have been generalized to codes over GF (q). A binary code of even length is said to be Z4 -linear if upto suitable permutation of coordinates it is the image of some linear code over Z4 . In[53], Hammons et al have shown that the first order Reed Müller code, Kerdock code and Preparata code are Z4 -linear. Since then many researchers have constructed various linear codes over Z4 and over Zps and studied their gray images (see[113],[15],[53],[21] etc.). In the present dissertation, we have constructed Simplex code of type α and β over Z4 and over Z2s . Some fundamental properties like 2-dimension, Hamming, Lee and Generalized Lee weight distributions, weight hierarchy etc. are determined for these codes. It is shown that binary images of these codes give rise to some interesting binary codes. The concept of chain condition defined for GF (q) is extended to codes over Z2s . In[110], Tiu and Wallace constructed a binary linear family of self orthogonal codes Cp based on the concept of quadratic residue and conjectured that dim(Cp ) = p. We have proved the conjecture and extended the construction to any odd prime p. It is shown that Cp is Z4 -linear. This dissertation is divided into seven chapters. Chapter 2 contains preliminary definitions and a brief survey of known results on codes over GF (q), and codes over integer modulo ps . Some new results are also included. Chapter 3 introduces Simplex codes of type α and type β, denoted Skα and Skβ over Z4 . Basic properties like 2- dimension, weight hierarchy, and Symmetrized weight enumerator etc. are obtained in section 1. The gray images of these codes are constructed in two ways and some interesting linear and nonlinear binary families are obtained. It 4 is also shown that both Skα and Skβ satisfy the chain condition. Chapter 4 deals with simplex codes of type α and β over Z2s . Using the generalized gray map introduced by Carlet, the result obtained in Chapter 3 over Z4 are extended for Simplex codes of type α and β over Z2s . The concept of chain condition for various self-dual and self-orthogonal codes over Z4 is studied in chapter 5. Section 1 deals with self dual codes of length at most 9 or of length 2m and 4m. A construction and some of the fundamental properties of the first order Reed Müller code over Z2s is given in section 2. In Chapter 6, we study binary linear Norm Quadratic Residue (NQR) codes Cp for any odd prime p. It is shown that Cp is Z4 linear, dim(Cp ) = p and its covering radius is p(p−1) . 2 The thesis concludes with some general remarks in the last chapter. Chapter 2 Preliminaries and Survey If you are faced by a difficulty or a controversy in science, an ounce of algebra is worth a ton of verbal argument. . . . J.B.S. Haldane (1892-1964) 2.1 Introduction A nonempty set S with two binary operations ‘+’ and ‘.’ ( called addition and multiplication) is a ring if (i) < S, + > is an abelian group, (ii) ‘.’ is associative , and (iii) distributive laws hold. If in addition, a.b = b.a for all a, b ∈ S, S is called a commutative ring. If S contains a multiplicative identity, denoted 1, then S is called a ring with unity. A nonzero element ‘a’ in a ring S with 1 is called a unit if there exists b ∈ S such that a.b = b.a = 1. A commutative ring with 1 in which every nonzero element is a unit is called a field. It is known that a finite field with q elements exists if and only if q is a prime power. A finite field having q elements is unique up to isomorphism, called the Galois field, denoted GF (q). Let q be a positive integer. The set Zq of all equivalent classes of integers modulo q, with respect to addition and multiplication modulo q, forms a commutative ring with unity. If p is a prime and s is a positive integer then the set of all units in Zps is 5 6 the set U = {ap + b|0 ≤ a ≤ p(s−1) − 1, 1 ≤ b ≤ p − 1}. The set of zero divisors in Zps is given by Z = {ap|0 ≤ a ≤ ps−1 − 1}. Let S be a commutative ring with unity. An abelian group < M, + > is called a left S-module if there exists a function µ : S × M → M denoted by µ(α, a) satisfying (i) µ(α, a + b) = µ(α, a) + µ(α, b), (ii) µ(αβ, a) = µ(α, µ(β, a)), (iii) µ(α + β, a) = µ(α, a) + µ(β, a), (iv) µ(1, a) = a. ∀α, β ∈ S and a, b ∈ M. The set S n = {x = (x1 , x2 , . . . , xn )|xi ∈ S} is a left S-module. A subset B of M is a generating set for M if every element of M is a finite linear combination of elements in B over S. B is called a basis for M if the above representation is unique for each element of M. A module which admits a basis is called a free module. If S is a field then a left S-module V is called a vector space. It is well known that every vector space has a basis. Moreover, if V has a finite basis then any two bases for V have the same number of elements. In such a case, if |B| = n, then V is called an n-dimensional vector space and we write, dim V = n. If S is a field then S n is an n-dimensional vector space over the field S. Let Fq be a set of q-alphabets. A code C of length n and size M is a subset of Fnq having M elements. The elements of C are called codewords. In order to be able to correct errors, if any, arising out of noise one would like to have some algebraic structure attached to the set Fq of alphabets. If q is a prime power one usually takes Fq to be GF (q) and otherwise Fq = Zq . Let Fq = GF (q)( Zq ) and let C ⊆ Fnq . If C is a subspace (submodule) of Fnq then C is called a linear code over Fq . In case Fq = GF (2)(GF (3)) then C is called a binary (ternary) code. Let x, y ∈ Fnq . The number of nonzero components in x is called the Hamming weight of x, denoted wH (x). Hamming distance dH (x, y) between x and y is the number of coordinates in which they differ. Note that for x, y ∈ C, dH (x, y) = wH (x − y). Let dH = min{dH (x, y) : x, y ∈ C, x 6= y}. dH is called the minimum Hamming distance 7 of C. A code C with minimum Hamming distance dH can correct up to ⌊ (dH2−1) ⌋ errors using nearest neighborhood decoding. Almost all codes in classical coding theory are defined for the Hamming distance. Hamming distance codes are ideal for the balanced channel in which error probability of each symbol is same. In 1958, C. Y. Lee, defined Lee distance which is suitable for memory less, discrete and symmetric channels[75]. Another useful distance is the Euclidean distance. The Lee (Euclidean) weight of an element a ∈ Zq is given by wL(a) = min {a, q − a} (wE (a) = (wL (a))2 ). The Lee (Euclidean) weight of a vector x ∈ Zqn is the sum of the Lee (Euclidean) weights of its components. For x, y ∈ Zqn , dL (x, y) = wL (x − y)(dE (x, y) = wE (x − y)) is called the Lee (Euclidean) distance between x and y. For binary codes the concept of Hamming, Lee and Euclidean distance coincide while for ternary codes the Hamming and Lee weight coincide. The minimum Lee (Euclidean) distance dL (dE ) of a code C is defined analogously. The choice of distance for any code depends on the decoding procedure and the channel under consideration. Let Fq = GF (q)( Zq ) and let C ⊆ Fnq . If C has M codewords and minimum Hamming, Lee and Euclidean distances as dH , dL and dE respectively then C is called an (n, M, dH , dL , dE ) code. If in addition, for Fq = GF (q), C is linear and dim C = k then C is called an [n, k, dH , dL, dE ] or simply an [n, k] code if we do not wish to specify distances. Any [n, k] linear code C, being a subspace can be specified either by a k × n generator matrix G (rows of G form a basis for C) or by an (n − k) × n full rank matrix H (called parity-check matrix) whose solution space is C. Therefore, C must contain a codeword of Hamming weight n − k + 1 and hence dH ≤ n − k + 1. (2.1) This bound is called the Singleton bound for Hamming distance. For given k and dH the minimum value of n for which there exists an [n, k, dH ] code is denoted by nq (k, dH ). & ' In 1965 Solomon and Stiffler[105] have shown that nq (k, dH ) ≥ Pk−1 i=0 dH qi ≡ gq (k, dH ). 8 The binary version of this was first proved by Griesmer in 1960[49]. A code meeting this bound is called a Griesmer code. An [nq (k, dH ), k, dH ] code is called an optimum code. In 1991, V. K. Wei introduced the notion of Generalized Hamming weights for binary linear codes[115]. This was later extended to linear codes over GF (q) by Klove et al[72] and to linear codes over Z4 by Ashikhmin[1]. It has been further studied by many researchers[60],[72],[59],[61],[22],[73]. Let C be an [n, k] linear code over GF (q) and let 1 ≤ r ≤ k. The r-th Generalized Hamming weight(GHW) , dr (C), of C is defined by dr (C) = min{wS (D) | D is an [n, r] subcode of C}, where wS (D), called the support size of D, is the number of coordinates in which some codeword of D has a nonzero entry. The set {d1 (C), d2 (C), . . . , dk (C)} is called the weight hierarchy of C. The following theorem summarizes some of the basic properties of the weight hierarchy of C. Theorem 2.1 Let C be an [n, k] linear code over GF (q). Then 1. (Monotonicity):1 ≤ d1 (C) < d2 (C) < · · · < dk (C) ≤ n 2. (Generalized Singleton Bound): dr (C) ≤ n − k + r; for 1 ≤ r ≤ k (r=1 yields (2.1)). 3. (Duality): {dr (C) : 1 ≤ r ≤ k} = {1, 2, . . . , n}\{n + 1 − dr (C ⊥ ) : 1 ≤ r ≤ n − k} In[60], Helleseth, Klove, Levenshtein and Ytrehus have proved the following lemma connecting the Hamming weight distribution and support size of any linear code C. Lemma 2.2 If D is an [n, r] linear code over GF (q) then X c∈D wH (c) = q r−1 (q − 1)wS (D). 9 Thus, an alternative definition of dr (C) is given by    X 1 wH (d) : Dr is an [n, r] subcode of C . dr (C) = r−1 min   q (q − 1) d∈Dr In another paper[116], Wei and Yang have introduced the concept of chain condition for studying the weight hierarchies of linear codes. A linear code C is said to satisfy the chain condition if there exists a chain D 1 ⊆ D2 ⊆ · · · ⊆ D k , of subcodes of C satisfying wS (Dr ) = dr (C), 1 ≤ r ≤ k. It is known that almost all good codes over GF (q) satisfy the chain condition[31],[23],[32],[24]. The weight hierarchies of linear codes over finite fields play an important role in Cryptography, for example, Wire-Tap Channel-II (WTC-II)[84]. For various other applications see[67],[56],[115] etc. In[2], Ashikhmin has shown that Octacode code over Z4 is better (from the point of view of GHW) than any optimal linear code over GF (4) of the same length and cardinality. Let x, y ∈ Fnq . The standard inner product of x and y is the scalar < x, y >= Pn i=1 xi yi in Fq . The subset C ⊥ = {x ∈ Fnq | < x, c >= 0, ∀c ∈ C} is called the dual code ⊥ of C. It is easy to verify that |C||C ⊥ | = |Fnq | and C ⊥ = C. If C is an [n, k] linear code over a field then C ⊥ is an [n, n − k] linear code. Moreover, G is a generator matrix of C if and only if, G is a parity-check matrix of C ⊥ . If C = C ⊥ (C ⊂ C ⊥ ) then C is called a self-dual (self orthogonal or weakly self dual) code. A self dual linear code C over a field must have even length. However this need not be true for codes over Z4 . In[87] Pless et al have constructed self dual codes of odd lengths over Z4 . A self dual code over Zq , q even, is called a code of type II(I) if all the Euclidean weights are divisible by 2q(q). For details on self dual codes, see the excellent survey written by Rains and Sloane[92]. 10 Two codes are said to be equivalent if one can be obtained from the other by applying finitely many operations of the type (i) permutation of coordinates, (ii) multiplying one or more coordinates by a unit. A mapping θ : C → C is called an automorphism of C if θ is 1 − 1, onto and a homomorphism. The collection of all automorphisms of C is a group called the automorphism group of C, denoted Aut(C). For each 1 ≤ i ≤ n, let AH (i)(AL (i)) be the number of codewords of Hamming (Lee) weight i in C. Then {AH (0), AH (1), . . . , AH (n)} ({AL (0), AL (1), . . . , AL (n)}) is called the Hamming (Lee) weight distribution of C. The Hamming weight enumerator (hwe) of C is a polynomial defined by WC (x, y) or HamC (x, y) = X xn−wH (c) y wH (c) = c∈C n X AH (i)xn−i y i . i=0 There is an analogous definition for nonlinear codes. Complete weight enumerator(cwe) for a code C over Zps is defined as follows. Let m = ps −1, and let c = (c1 , c2 , . . . , cn ) ∈ C. For each j ∈ Zps , let ωj (c) = |{k : ck = j}|. Then the cwe is the polynomial cweC (x0 , . . . , xm ) = P c∈C x0 ω0 (c) x1 ω1 (c) · · · xm ωm (c) . Note that permutation equivalent codes have the same cwe but in general two equivalent codes may have different cwe’s. The appropriate weight enumerator for an equivalent class of codes is called a Symmetrized weight enumerator(swe). For example, the swe of a code over Z4 is the polynomial: swe(x, y, z) = cwe(x, y, z, y). The Lee weight enumerator of C is the homogeneous polynomial of degree 2n given by LeeC (x, y) = P c∈C x2n−wL (c) y wL(c) . The Hamming weight enumerators of code C and C ⊥ are connected by the identity HamC ⊥ = 1 HamC (x + y, x − y) |C| (2.2) called the MacWilliams identities. Similar identities have been obtained for codes over rings[92]. The following analogue of (2.2) holds for codes over Z4 . sweC ⊥ (x, y, z) = LeeC ⊥ (x, y) = 1 sweC (x |C| 1 LeeC (x |C| HamC ⊥ (x, y) = + 2y + z, x − z, x − 2y + z). + y, x − y). 1 HamC (x |C| + 3y, x − y). 11 2.2 Classical Codes over GF (q) Let Fq = GF (q) = {0, 1, α3 , . . . , αq }. For a given k and q, let Gk (q) be a k × (q k − 1)/(q − 1) matrix over Fq in which any two columns are linearly independent. The code Sk (q) generated by the matrix Gk (q) is called the Simplex code. Note that i h Sk (q) is a (q k − 1)/(q − 1), k, q k−1 code. It is known that any linear code with the above parameters is equivalent to Sk (q)[40]. Gk (q) can be defined inductively by   G2 (q) =  and   Gk (q) =   00 · · · 0  0 1 1 α3 · · · αq−1 αq  1 0 1 1 1 11 · · · 1 ··· 1 1 ,  α3 · · · α3 · · · αq · · · αq  Gk−1 (q) 0 Gk−1 (q) Gk−1 (q) · · · Gk−1 (q) .  Every nonzero codeword of Sk (q) has weight q k−1 . The binary simplex code (usually denoted Sk ) was first discovered by Ronald A. Fisher[35] in 1942 in connection with statistical designs. In 1945, it was further generalized to arbitrary prime powers[36]. The dual of the Simplex code is the well known h q k −1 q k −1 , q−1 q−1 i − k, 3 Hamming Code. A nice way to construct new codes is by puncturing or appending one or more coordinates of a known code. Let 1 ≤ u ≤ k − 1 and let Gk,u (q) be the matrix obtained from Gk (q) by deleting columns corresponding to the columns of the matrix Gu (q), i.e, Gk,u (q) = where 0 is a (k − u) × q u −1 q−1  Gk (q) \ 0 Gu (q)  , zero matrix and [A\B] denotes the matrix obtained from the matrix A by deleting the columns of the matrix B. The code Mk,u (q) generated by the matrix Gk,u (q) is the punctured code of Sk (q) and is called a MacDonald code. Mk,u (q) is a h q k −q u , k, q k−1 q−1 i − q u−1 code in which every nonzero codeword has weight either q k−1 or q k−1 − q u−1 [77]. 12 h In 1954, I. S. Reed and R. Müller constructed a 2m , Pr i=0   m i i , 2m−r binary lin- ear code, denoted RM(r, m), called the r th -order Reed Müller code for all m ≥ 0, 0 ≤ r ≤ m. If G(r, m) is a generator matrix for RM(r, m) then   G(r + 1, m + 1) =   G(r + 1, m) G(r + 1, m)  0 G(r, m)  is a generator matrix for RM(r + 1, m + 1). Further RM(r, m)⊥ = RM(m − r − 1, m). Let A(n, dH ) = max{M| there exists a binary (n, M, dH ) code}. It is known that A(16, 6) ≤ 28 = 256. Hence a question arose whether there exists a (16, 256, 6) code or not? In[83], Nordstrom constructed a (16, 256, 6) nonlinear code popularly known as Nordstrom Robinson code. In 1968, Preparata generalized this by constructing a m −2m) (2m , 2(2 , 6) binary nonlinear code, P (m), called Preparata code for m even and  m ≥ 4 [89]. Another construction of a 2m , 22m , 2m−1 − 2(m−2)/2  binary nonlinear code K(m), called Kerdock code for m even and m ≥ 4 was given by Kerdock in 1972 [70]. For m = 4, P (4) = K(4). Further nonlinear generalizations were later found by Goethals, Delsarte and Hergert[28],[42],[43],[62]. 2.3 Codes over Integer Residue Rings Codes over Zq have various applications including phase modulated channels[6],[81], multifrequency phase telegraphy[82], multilevel quantized pulse amplitude modulated channels[6], multilevel transmission system[80], multiplexing in multiaccess communication system[109], multichannel communication[109]etc. By Chinese Remainder theorem, the study of codes over Zq can be reduced to the study of codes over Zps . Many different kind of codes over Zq have been constructed in seventies by Blake[7],[8], Spigel(1977)[106], Priti Shankar (1979)[100] etc. In early eighties not much attention was paid to such codes[114]. This is partly due to the difficulties arising due to the presence of zero divisors in Zps . In 1989, P. Solé[104] discovered a family of nearly optimal 13 four-phase sequences of period 22r+1 − 1 with alphabet 1, i, −1, −i, where i = √ −1. This was also found independently by Boztas, Hammons and Kumar in 1990 [13]. The above family may be viewed as a linear code over Z4 by identifying the alphabet ia by its exponent a. When studying these four-phase sequences, Hammons and Kumar and later independently Calderbank, Sloane and Solé noticed the resemblance between the 2-base expansion of the quaternary codewords and the standard construction of the Kerdock codes. Thus they realized that the Kerdock code is simply the image of a linear code over Z4 under a suitable map. In October 1992, in a DIMACS/IEEE workshop M. D. Trott, acting on a suggestion by Forney proved that the NordstromRobinson (NR) code is the binary image of a linear code over Z4 , called Octacode[38]. However, this was already known to Hammons and Kumar in June 1992 [13]. After realizing that they are working on the same problem they formed a group. Combined efforts of the group lead to an award winning paper[53] in which they could map various good nonlinear binary families of codes to some linear codes over Z4 . They have shown that Kerdock and Preparata codes are actually dual to each other when viewed as a linear code over Z4 . They observed that ( Z4n , Lee distance) and ( Z22n , Hamming distance) are isometric to each other and hence obtained the explanation for the m mysterious connection |P (m)||K(m)| = 22 . The mapping φ : Z4 → Z22 defined by φ(0) = 00, φ(1) = 01, φ(2) = 11 and φ(3) = 10 is called the Gray map. It is extended from Z4n → Z22n by applying φ to each coordinate and is popularly known as Gray map. It was used by F. Gray to avoid large errors in transmitting signals by pulse code modulation (PCM) in 1940’s. Clearly, Gray map is distance preserving[53]. If C is a linear code over Z4 then φ(C) will denote the image of C under the Gray map. C is called Z2 -Linear if φ(C) is a binary linear code. A binary code is said to be Z4 -linear if it is equivalent to φ(C) for some linear code C over Z4 . A necessary and 14 sufficient conditions for Z4 -linearity ( Z2 -linearity) is given by the following Theorem. Theorem 2.3 Hammons et al[53] 1. A binary linear code C of even length is Z4 -linear if and only if its coordinates can be permuted so that u, v ∈ C ⇒ (u + σ(u)) ⋆ (v + σ(v)) ∈ C (2.3) where σ is the swap map that interchanges the left and the right halfs of a vector and ⋆ denotes the componentwise product of two vectors. 2. For each a ∈ Z4 , let ā be the reduction of a modulo 2 and let C be a linear code over Z4 , then C is Z2 -linear if and only if c = (c1 , . . . , cn ) and c′ = (c′1 , . . . , c′n ) ∈ C implies 2c̄ ⋆ c̄′ = (2c¯1 c¯1 ′ , . . . , 2c¯n c¯n ′ ) ∈ C. It is known that the r th order binary Reed Müller code R(r, m) is Z4 -linear for r = 0, 1, 2, m − 1 and m[53] and is not Z4 - linear for other values of r[63]. The Golay code and the extended Hamming code of length n = 2m (m ≥ 5) are not Z4 -linear[53]. Let h2 (x) ∈ Z2 [x] be an irreducible polynomial over Z2 such that h2 (x)|(xn −1) over Z2 . Then Hensel lemma ( see[78]) guarantees the existence of a polynomial h(x) ∈ Z4 [x] such that (i) h(x) ≡ h2 (x) (mod 2), (ii) h(x)|(xn − 1) over Z4 . This polynomial h(x) can be obtained, for example, by Graffe’s method, called the Hensel uplift of h2 (x)[11]. In[53], Hammons et al have shown that the gray image of the Hensel uplift of the first order Reed Müller code R(1, m) is the Kerdock code and the Gray image of the Hensel uplift of the extended Hamming code is Preparata code (differ slightly from classical Preparata code but shares the same distance structure). Thus these nonlinear codes are extended cyclic codes over Z4 . This process of Hensel uplifts can be continued till Z2s or in general up to the ring of 2-adic integers. The classical Theory of cyclic codes over finite fields was extended to cyclic codes over Z4 by Pless and Quian[88] and over Zps by Pramod Kanwar and Permouth[68]. 15 Some basic theorems giving the structure of cyclic codes of length n over Zps and over the p-adic numbers was considered by Calderbank and Sloane[15]. A generalization of McEliece theorem that characterizes the possible Hamming weights of a binary cyclic code was given by Calderbank, Li and Poonen[17]. 2.4 Codes over Zps Let C be a linear code of length n over Zps . Then C is a finite abelian group of type psk0 p(s−1)k1 · · · pks−1 with Ps i=0 ki = n and ki (0 ≤ i ≤ s) are nonnegative integers[15]. Using this Calderbank and Sloane[15] have shown that C has a generator matrix G (rows of G generate C) of the form  G= where Aij Ik0 0 A01 ··· A0s−1 A0s pIk1 pA12 · · · pA1s−1 pA1s p2 Ik2 · · · .. .. . . p2 A2s−1 .. . p2 A2s .. . 0 .. . 0 .. . 0 0 are matrices over blocks of size k0 , |C| = (ps )               Ps−1 i=0 ki k1 ,  ··· , A02 · · · ps−1 Iks−1 ps−1 As−1s 0 ks−1 , k is a free module if and only if ki = 0 ks . Let k ks−1  = Ps−1 i=0 (s H= (2.4) − i)ki . which is equal to pk . for all ··· B0s B0s−1 pB1s .. . pB1s−1 · · · pIks−1 .. .. .. . . . ps−1 Bs−1s ps−1 Ik1 · · · Then Note that C i = 1, 2, · · · , s − 1. It is easy to see that a generator matrix for C ⊥ is of the form                  ,       and the columns are grouped into Zps / 1k0 pk1 (p2 ) 2 · · · (ps−1 )   B01 0  Iks   0 .. . 0    ,     where Bij are matrices over Zps and the columns are grouped into blocks of size ks , ks−1, · · · , k1 . Therefore |C ⊥ | = (ps ) Ps−1 i=0 ki  / 1ks pks−1 (p2 ) ks−2 · · · (ps−1 ) k1  = psn−k . 16 Note that sn − k = 2.4.1 Ps i=1 iki . p-dimension of Linear Codes over Zps The presence of zero divisors in Zps creates problem in defining linear dependence of vectors in Zpns . For example, the following two statements are equivalent for a subset D of a vector space over a field. 1. A nontrivial linear combination of vectors in D is zero. 2. One of the vector in D is a linear combination of some other vectors of D. However, these are not equivalent for modules over Zps . For example, the subset D = {(1, 2), (1, 0)} of Z42 satisfies 1 but not 2. Also, note that the module generated by (2,0) and (0,2) over Z4 are properly contained in the module generated by (1,0) and (0,1). Consequently, defining the dimension of a module as a cardinality of its basis is not meaningful. Recently, while studying Trellis description of linear codes over Zps Vazirani, Saran and Sundar Rajan[112] have introduced the following notion of p-dimension for finitely generated modules over Zps . A vector v ∈ Zpns is a p-linear combination of the vectors v1 , v2 , · · · , vk if v = Pk i=1 λi vi with λi ∈ Zp ; 1 ≤ i ≤ k. Let D = {v1 , v2 , · · · , vk } be an ordered subset of Zpns . Then p-span of D is the set nP k i=1 o ai vi : ai ∈ Zp . D is called a p-generating sequence if for each i = 1, 2, · · · , k − 1, pvi is a p-linear combination of vi+1 , ..., vk and pvk = 0. If in addition, 0 is a nontrivial p-linear combination of v1 , v2 , · · · , vk then D is called p-linearly dependent. A subset B of C is a p-basis for the linear code C over Zps if B is p-linearly independent and C is the p-span of B. The number of vectors in any two p-bases for C is same and is called p-dimension of C, denoted p−dim(C). The following theorem summarizes results for a p-basis. Theorem 2.4 [112] 17 1. Every module over Zps admits a p-basis.   2. p−dim Zpns = sn. 3. B is a p-basis for C if and only if every vector in C is a unique p-linear combination of vectors in B. Let C be the linear code generated by the matrix B given in (2.4) and B be the matrix  Ik0     pIk0   ..  .    s−1  p Ik0     0    0    ..  .  B=  0     0     0   ..  .     0   ..   .   0 A01 A02 ··· A0s−1 A0s pA01 .. . pA02 .. . ··· .. . pA0s−1 .. . pA0s .. . ps−1 A0s−1 ps−1 A0s ps−1 A01 ps−1 A02 · · · pIk1 pA12 ··· pA1s−1 pA1s p2 Ik1 .. . p2 A12 .. . ··· .. . p2 A1s−1 .. . p2 A1s .. . ps−1 A1s−1 ps−1 A1s ps−1 Ik1 ps−1 A12 · · · 0 p2 Ik2 ··· ··· p2 A2s 0 .. . p3 Ik2 .. . ··· .. . ··· .. . p3 A2s .. . 0 .. . ps−1 Ik2 .. . ··· .. . ··· .. . ps−1 A2s .. . 0 0 ··· ps−1 Iks−1 ps−1 As−1s                         .                       (2.5) Then for an appropriate permutation matrix P it is easy to see that rows of P B form a p-basis for the code generated by B. Hence p −dim(C) = k = Ps−1 i=0 (s − i)ki . Note that, for an [n, k] linear code C over Zps , the dual code C ⊥ is an [n, sn − k] linear code. The p-span of the vectors v1 , . . . , vk will be denoted by < v1 , . . . , vk > . 18 2.4.2 Generalized Gray Map It was observed in section 2.3 that the image of the linear code over Z4 under the Gray map is a binary code of twice the length and whose minimum Hamming distance is same as the minimum Lee distance of C. Thus, it is natural to ask: Does there exist a generalized gray map which maps a linear code over Zps to a code over Zp having similar properties? Unfortunately, no such generalization is known. Infact, recently, Ana Sălăgean- Mandache[98] has shown that except for the well-known case p = s = 2, it is not possible to construct a weight function on Zps for which Zps is isometric to Zps with the Hamming metric. In[19], C. Carlet has defined a generalized gray map φG s−1 from Z2s to Z22 and has obtained the Z2s version of Kerdock and Delsarte-Goethals codes. Let s be a positive integer and let Es be the s × 2s matrix whose columns consist of binary representation of numbers 0, 1, 2, · · · , 2s − 1 (elements of Z2s ). Thus Es can be inductively defined by   Es =   Es−1  Es−1  00 . . . 0 11 . . . 1 ,  with E1 = [01]. Let s > 1 and let u ∈ Z2s , 0 ≤ u < 2s . Let u = Ps i=1 (2.6) ui 2i−1 (ui = 0 or 1) and let s−1 Es−1 = [yij ]; 1 ≤ i ≤ s − 1, 0 ≤ j ≤ 2s−1 − 1. Define a map φG : Z2s 7→ Z22 by φG (u) = (x0 , x1 , . . . , x2s−1 −1 ); where xj = us + s−1 X i=1 ui yij ∈ Z2 . (2.7) Note that φG is a mapping from Z2s onto the Reed Müller code of order 1 and length 2s−1 . φG can also be defined as a Boolean function φG (u) evaluated on GF (2s−1 ) by : φG (u) : (y1 , y2 , · · · , ys−1) 7→ us + s−1 X ui yi . i=1 φG is a mapping from Z2s onto the Reed Müller code of order 1 and length 2s−1 . 19 Note that this map is distance preserving[19]. For s = 3 this is a well understood generalization of the gray map but for s > 3 still several things need to be investigated. Note that Es−1 is the generator matrix of the extended binary simplex code of dimension s − 1. Thus weight of any row will be 2s−2 . Hence corresponding to entries 1, 2, 22, · · · , 2s−2 of Z2s , φG (u) will have weight 2s−2 and the weight of φG (2s−1 ) will be 2s−1 (see (2.6) and (2.7)). Now all other nonzero elements of Z2s will also give the weight 2s−2 since the weight of other entries corresponds to some linear combination of rows of Es−1 . Thus, for u 6= 0, we have wt(φG (u)) =      2s−2, u 6= 2s−1 (2.8) 2s−1, u = 2s−1 . We call this weight the Generalized Lee weight of u and is denoted by wGL (u). Also, we have the matrix  φG (1)    φ (2)  G    φG (22 )  Gs (say) =  ..   .    φ (2s−2 )  G  φG (2s−1 )          =         0101 0101 0101 0101 · · ·    0011     0000   .  .  .    0000   0011 0011 0011 · · · 1111 0000 1111 · · · .. .. .. .. . . . . ··· 0000 ··· ··· 1111 1111 1111 1111 · · ·  0101 0101 0101   0011 0011 0011   1111 0000 .. .. . . 1111 ···   1111   . ..   .   (2.9)  1111    1111 1111 1111 Also, let Gs−i denote the matrix obtained from Gs by dropping first i rows. Note that φG can be naturally extended to Z2ns by applying φG to its components. Definition 2.1 A binary code is called Z2s -linear if it is equivalent to φG (C) for some linear code C over Z2s . A necessary and sufficient condition for the Z23 -linearity is given in[19]. It is also shown that any Z2s -linear code is distance invariant and dH (φG (u), φG (v)) = wGL (u − v)[19]. The minimum generalized Lee weight, dGL , of C can be defined in the usual sense. Note also that for s = 2, dGL = dL . The following Lemma is useful in Sections 3.3 and 4.2.1. 20 Lemma 2.5 For p = 2, the images of the rows of the matrix B given by (2.5) under the generalized gray map φG are linearly independent over Z2 . Proof: Applying the generalized gray map φG to the rows of B with a suitable arrangements of rows yields a binary matrix of the type    Gs  0 .. 0 0 Gs 0 0 0 Gs−1 0 0 0 0 .. 0 .. . 0 .. . 0 .. . 0 .. . 0 .. . 0 0 0 0 0 0 0 G2 0 0 0 0 0 0 0 0 .. 0 0 0 0 0 0 0 0 0 G2 0 0 0 0 0 0 0 0 0 0                                    . . Gs−1 .. .. . . . G1 ∗                   ,                   where Gs and Gs−i are as defined in (2.9) and ∗ is some block. It is easy to see that the rows of the above matrix are linearly independent. Remark 2.1 Lemma 2.5 implies that the transform of a 2-basis of C under the generalized gray map gives rise to linearly independent over Z2 . The converse of this need not be true, in general, For example, If s = 2, then {101100,011100} are Z2 -independent but {320,120} are not. A linear code C over Z2s of length n, 2-dimension k, minimum Hamming distance dH , minimum Lee distance dL and minimum Generalized Lee distance dGL is called an [n, k, dH , dL , dGL] code or simply an [n, k] code. C is called Z2 -linear if φG (C) is a 21 binary linear code. Thus, if C is an [n, k, dH , dL, dGL ] Z2 -linear code then φG (C) is a binary linear code of length 2s−1 n, dimension k and minimum Hamming distance dGL . Hence, by the Griesmer bound for binary linear codes[79], we have & & X dGL 1 k−1 n ≥ s−1 2 2i i=0 2.4.3 '' . (2.10) Generalized Hamming weights of Linear codes over Zps Let C be an [n, k] linear code over Zps . The definitions of Generalized Hamming weight (GHW), weight hierarchies and chain condition given in section 2.1 for linear codes over GF (q) can be extended in a similar fashion for linear codes over Zps . The only difference is that we replace dimension by p-dimension everywhere. Thus for 1 ≤ r ≤ k, the r th generalized hamming weight of C is defined as dr (C) = min{wS (Dr ) : Dr is an [n, r] subcode of C}, where wS (Dr ) is the support size of a subcode Dr of C with p−dim(Dr ) = r. The following theorem summarizes basic properties of GHW[2]. Theorem 2.6 Let C be an [n, k] linear code over Zps . Then 1. (Monotonicity): 1 ≤ d1 (C) ≤ d2 (C) ≤ · · · ≤ dk (C) ≤ n and if, dr (C) = dr+1 (C) = · · · = dr+s−1(C), then dr+s−1(C) < dr+s (C). 2. (Duality): {dr (C) : 1 ≤ r ≤ k} = {1, · · · , 1, 2, · · · , 2, . . . , n, · · · , n}\{n + 1 − dr (C ⊥ ) : 1 ≤ r ≤ sn − k} | {z s } | {z s } | {z For p = 2, the following Lemma is a generalization of Lemma 2.2. Lemma 2.7 If D is an [n, r] linear code over Z2s then 1. 2. P P c∈D wL (c) = 2r+s−2wS (D), c∈D wGL (c) = 2r+s−2wS (D). s } 22 Proof: Consider the (2r × n) array of all codewords in D. Let 0 ≤ m ≤ s and let nm−1 be the number of columns that contain the entries 0, 1 · 2s−m, 2 · 2s−m , 3 · 2s−m , · · · , (2m − 1) · 2s−m equally often. Then X wL (c) = m=1 c∈D = s X s  X m=1  r nm−1  2  2m Ps m=1 nm−1 = wS (D) and hence  (2m −1) X t=0 wL (t · 2s−m ) (2.11)  nm−1 2r−m (2s+m−2 ) = 2r+s−2 · wS (D) as wL (t · 2 s−m )=      t · 2s−m , t 6= 2m−1 2s−1 , otherwise. This proves 1. Proof of 2 follows using (2.8) and an equation similar to the equation (2.11). Remark 2.2 For s = 2, above Lemma was first proved by K. Yang et al (cf.[117]). Corollary 2.8 (Plotkin-type Bound) Let C be an [n, k, dL , dGL] linear code over Z2s . Then dL ≤ n · 2s−1 and dGL ≤ n · 2s−1 . Proof: n· 2k+s−2 (2k−1 ) By Lemma 2.7, the average Lee weight of the codewords will be ≥ dL . The proof of the following Corollary follows immediately by Lemma 2.7. Corollary 2.9 If 1 ≤ r ≤ k, then r th GHW of C satisfies & ' & ' (2r − 1)dL (2r − 1)dGL dr (C) ≥ , and d (C) ≥ . r 2r+s−2 2r+s−2 Remark 2.3 In view of Lemma 2.7, GHW can be defined alternately as dr (C) = 1 2r+s−2 min  X  d∈Dr wL (d) : Dr is an [n,r] subcode of C    for 1 ≤ r ≤ k. 23 Similar definition hold for wGL . If r = 1, we get the following corollary from Lemma 2.7. In[93](see also[25]), Rains has proved that for a linear code over Z4 , dH ≥ ⌈ d2L ⌉. The following corollary generalizes it. Corollary 2.10 Let C be a Linear code over Z2s , then ' & & ' dGL and dH ≥ s−1 . 2 dL dH ≥ s−1 , 2 A linear code over C over Z2s is said to be of type α (β) if & dGL dH = s−1 2 2.4.4 ' & dGL dH > s−1 2 '! . Codes over Z4 Codes over Z4 have been studied extensively (cf.[103],[113] etc.). In this section we collect some basic definitions specific to these codes and some further results. A linear code C over Z4 has a generator matrix G of the form  A  Ik0 G= 0 2Ik1  B1 + 2B2  , 2C where A, B1 , B2 and C are matrices with entries 0 and 1 and Ik is the identity matrix of order k. One can associate two binary linear codes with C as follows. The residue code , C (1) of C is given by C (1) = {(c¯1 , c¯2 , . . . , c¯n ) : (c1 , c2 , . . . , cn ) ∈ C} where c¯i denotes the reduction of ci modulo 2. Another binary linear code C (2) , called the torsion code of C is given by C (2) =  c : c = (c1 , . . . , cn ) ∈ C and ci ≡ 0 2  (mod 2) f or 1 ≤ i ≤ n . If k1 = 0 then C (1) = C (2) . The generator matrices of these codes are given by G(1) and G(2) , respectively, where G(1) =  Ik0 A B1    Ik0 and G(2) =  0 A Ik1  B1  C . 24 ⊥ If C is self orthogonal then C (1) is doubly even and C (1) ⊂ C (2) ⊂ C (1) and if C is self ⊥ dual then C (2) = C (1) [26]. Let c = (c1 , c2 , . . . , cn ) ∈ C. The Correlation of c is defined by θ(c) = (ω0 (c) − √ ω2 (c)) + i(ω1 (c) − ω3 (c)); where i = −1 and ωj (c) = |{k : ck = j}| [104]. If d1 , d2 , . . . , dk is the weight hierarchy of an [n, k, dH , dL ] linear code C over Z4 then for s = 2, and p = 2 by Theorem 2.6 and Corollary 2.10, Hence n − dL 2 ≥ k−1 . 2 dL 2 ≤ d1 < d3 < · · · < dk ≤ n. Thus we get the following corollary. Corollary 2.11 For any [n, k, dH , dL] linear code over Z4 , n ≥ l dL +k−1 2 m . If C is Z2 -linear then the following proposition gives another lower bound, though not tight for type β code of C. Proposition 2.1 Let C be a Z2 -linear code over Z4 . Then & & X dL 1 r−1 dr (C) ≥ 2 i=0 2i Proof: '' . Follows easily using inequality (2.10). The following upper bound on the weight hierarchy of a linear code C follows from Theorem 2.6 for s = 2. Proposition 2.2 1. dr (C) ≥ n − 2. dr (C) ≥ n − (k−r−1) ; (k 2 3. dr (C) ≥ n − (k−r−2) ; (k, r 2 (k−r) ; (k, r 2 even ). even , r odd or k odd, r even ). odd ). Chapter 3 Z4-Simplex Codes and their Gray Images The binary Gray code is fun, For in it strange things can be done, Fifteen, as you know, Is one,oh,oh,oh, And ten is one,one,one, and one. . . . Anon 3.1 Introduction In this chapter, we introduce simplex codes of types α and β over Z4 . These are generalizations of binary simplex codes in many ways. The torsion/reduction codes of these families are equivalent to several copies of binary simplex codes. Basic properties of these codes are determined in section 1. The gray images of these codes are constructed in two ways in section 2. It is shown that these families satisfy the chain condition. Generalization of these codes to codes over Z2s is done in the next chapter. 25 26 3.2 Z4-Simplex Codes of Type α and β Let Gk be a k × 22k matrix over Z4 defined inductively by   Gk =    00 · · · 0 11 · · · 1 22 · · · 2 33 · · · 3  Gk−1 Gk−1 Gk−1 (3.1)   Gk−1 with G1 =[0123]. Note that columns of Gk consist of all distinct k-tuples over Z4 . Clearly, the code Skα generated by Gk has length 22k and 2-dimension 2k. The following observations are useful in determining Hamming and Lee weights of Skα . α Remark 3.1 If Ak−1 denotes an array of codewords in Sk−1 and if i = (iii...i) then an array of all codewords of Skα is given by   Ak−1          Ak−1 Ak−1 Ak−1 Ak−1 1 + Ak−1 2 + Ak−1 3 + Ak−1 Ak−1 2 + Ak−1 Ak−1 2 + Ak−1 Ak−1 3 + Ak−1 2 + Ak−1 1 + Ak−1       .     Remark 3.2 If R1 , R2 , ..., Rk denote the rows of the matrix Gk then wH (Ri ) = 3 · 22k−2 , wH (2Ri ) = 22k−1 and wL (Ri ) = 22k = wL (2Ri ). It may be observed that each element of Z4 occurs equally often in every row of Gk . In fact we have the following lemma. Lemma 3.1 Let c(6= 0) ∈ Skα . If one of the coordinates of c is a unit then every element of Z4 occurs 4k−1 times as a coordinate of c. Otherwise wH (c) = 22k−1 . Proof: y1 =  α By Remark 3.1, any x ∈ Sk−1 gives rise to following four codewords of Skα x x x x  , y2 =  x 1+x 2+x 3+x  , 27 y3 =  x 2+x x 2+x  and y4 =  x 3+x 2+x 1+x Hence by induction, the assertion follows.  . Let G(Sk ) (columns consist of all nonzero binary k-tuples) be a generator matrix for an [n, k] binary simplex code Sk . Then the extended binary simplex code Sˆk is generated by the matrix G(Sˆk ) =  0 G(Sk )  . Inductively,   G(Sˆk ) =   00 . . . 0  11 . . . 1  ˆ ) G(Sk−1 ˆ ) G(Sk−1   with G(Sˆ1 ) = [0 1] . (3.2) Lemma 3.2 Torsion code of Skα is equivalent to the 2k copies of Sˆk . Proof: Recall that torsion code of Skα is the set of codewords obtained by replacing 2 by 1 in all 2-linear combination of the rows of the matrix 2Gk where Gk is defined in (3.1). The proof is by induction on k. It is easy to see that the result holds for k = 2. ˆ ) then the matrix 2Gk If 2Gk−1 is permutation equivalent to the 2k−1 copies of 2G(Sk−1 takes the form " 00 · · · 0 22 · · · 2 00 · · · 0 ˆ ) · · · 2G(Sk−1 ˆ ) 2G(Sk−1 ˆ ) · · · 2G(Sk−1 ˆ ) 2G(Sk−1 ˆ ) · · · 2G(Sk−1 ˆ ) 2G(Sk−1 22 · · · 2 ˆ ) ··· 2G(Sk−1 ˆ ) 2G(Sk−1 Regrouping the columns according to (3.2) (Infact, 2G(Sˆk )) gives us the desired result. As a consequence of Lemmas 3.1 and 3.2 we determine Hamming and Lee weight distributions of Skα . Theorem 3.3 Hamming and Lee weight distribution of Skα are: 1. AH (0) = 1, AH (22k−1 ) = 2k − 1, and AH (3 · 22k−2 ) = 2k (2k − 1) # . 28 2. AL (0) = 1, AL (22k ) = 22k − 1, Proof: By Lemma 3.1, each nonzero codeword of Skα has Hamming weight either 3 · 4k−1 or 22k−1 and Lee weight 22k . By Lemma 3.2, the dimension of the torsion code of Skα is k, Thus there will be 2k − 1 codewords of Hamming weight 22k−1 . Hence the number of codewords having Hamming weight , 3 · 4k−1 , is 4k − 2k . Remark 3.3 1. Skα is an equidistant code with respect to Lee distance whereas Sk is an equidistant binary code with respect to Hamming distance. 2. Skα is of type α. 3. Skα is a 4-ary balanced code (see[111]). 4. The minimum Euclidean weight dE of Skα is 3 · 22k−1 . Let S¯kα be the punctured code of Skα obtained by deleting the zero coordinate. Then the swe (see section 2.4.4) of S¯kα is 2k−1 swe(x, y, z) = xn(k) + (2k − 1)(xz)n(k−1) z[(xz)n(k−1)+1 + 2k y 2 ], where n(k) = 4k − 1 and correlation of any c ∈ S¯kα is given by θ(c) = −1. (3.3) The length of Skα is large compared to its 2-dimension and increases fast with increment in 2-dimension. But one can always puncture some columns from Gk to yield good codes over Z4 in the sense of having maximum possible Lee weights for a given length and 2-dimension. Let Gβk be the k × 2k−1(2k − 1) matrix defined inductively by   Gβ2 =    1111 0 2  0123 1 1 ,  (3.4) 29 and for k > 2   Gβk =    11 · · · 1 00 · · · 0 22 · · · 2  Gk−1 Gβk−1 ,  Gβk−1 (3.5) α where Gk−1 is the generator matrix of Sk−1 . Note that Gβk is obtained from Gk by deleting 2k−1 (2k + 1) columns. By induction it is easy to verify that no two columns of Gβk are multiple of each other. Let Skβ be the code generated by Gβk . Note that Skβ is a [2k−1(2k − 1), 2k] code. To determine Hamming (Lee) weight distributions of Skβ we first make few observations. β α Remark 3.4 If Ak−1 (Bk−1) denotes an array of codewords in Sk−1 (Sk−1 ) and if i = (i, i, . . . , i) then an array of all codewords of Skβ is given by            Ak−1 Bk−1 Bk−1 1 + Ak−1 Bk−1 2 + Bk−1 2 + Ak−1 Bk−1 Bk−1 3 + Ak−1 Bk−1 2 + Bk−1       .     Remark 3.5 Each row of Gβk has Hamming weight 2k−3[3(2k − 1) + 1] and Lee weight 2k−1 (2k − 1). Proposition 3.1 Each row of Gβk contains 22(k−1) units and ω0 = ω2 = 2k−2(2k−1 −1). Proof: The result can be easily verified for the k = 2. Assume that the result holds for each row of Gβk−1 . Then the number of units in each row of Gβk−1 is 22(k−2) . By Lemma 3.1, the number of units in any row of Gk−1 is 22k−3 . Hence the total number of units in any row of Gβk will be 22k−3 + 2 · 22(k−2) = 22(k−1) . A similar argument holds for the number of 0’s and 2’s. Infact, similar to Skα , we have the following lemma 30 Lemma 3.4 Let c ∈ Skβ , c 6= 0. If one of the coordinates of c is a unit then ω1 (c) + ω3 (c) = 22(k−1) and ω0 (c) = ω2 (c) = 2k−2(2k−1 − 1). Otherwise ω1 (c) = ω3 (c) = 0, and ω0 (c) = 2k−1 (2k−1 − 1), ω2 (c) = 22(k−1) . Proof: β α By Remark 3.4, there exist y1 ∈ Sk−1 and y2 ∈ Sk−1 such that c can have any of the following four forms: c=  c=  y1 y2 y2  c=  . . . (iii) and c =  , . . . (i) 2 + y1 y2 y2  1 + y1 y2 2 + y2  , . . . (ii) 3 + y1 y2 2 + y2  . . . (iv). We prove the result by induction on k. Clearly result holds for k = 2. So assume β β that it holds for Sk−1 . Let y2 ∈ Sk−1 such that y2 is neither all zero nor all 2 codeword. If all the components of c(6= 0) are zero divisors (Note that this can happen only in β the first or the third form of c). In either of these forms of c, if y2 ∈ Sk−1 then by α induction hypothesis ω0 (y2 ) = 2k−2 (2k−2 − 1) and ω2 (y2 ) = 22(k−2) . Since y1 ∈ Sk−1 , by Lemma 3.1, ω0 (y1 ) = 22k−3 = ω2 (y1 ). Thus if c is of the form (i) and then ω0 (c) = 2{2k−2(2k−2 − 1)} + 22k−3 = 2k−1 (2k−1 − 1) and ω2 (c) = 2{22(k−2) } + 22k−3 = 22(k−1) so the result holds. Similar arguments hold if c is of the form (iii). If one of the coordinate of c is a unit then c can be of any form (i)-(iv) above. In any of these forms by Lemma 3.1 each entry in (i + y1 ); will occur equally often (= 4k−2 times). Therefore total number of units in c will be 2 × 4k−2 + 22(k−2) + 22(k−2) = 22(k−1) Similarly, the number of zero divisors occurs equally often 2k−2(2k−1 − 1) times. The case when y2 = 0 or 2 is trivial as codewords will be the multiple of the first row of Gβk only. Lemma 3.5 The torsion code of Skβ is equivalent to the 2k−1 copies of the binary simplex code Sk . Proof: The proof is by induction on k and is similar to that of Lemma 3.2. If 2Gβk−1 is permutation equivalent to the 2k−2 copies of 2G(Sk−1) then 2Gβk , where Gβk is given 31 by (3.5), takes the form   2Gβk =    22 · · · 2 22 · · · 2 00 · · · 0 0 2G(Sk−1) · · · 2G(Sk−1 ) 2G(Sk−1) · · · 2G(Sk−1)  .  Now grouping one column from first block together with one block of 2G(Sk−1 ) from each of second and third block yields the desired result. Again, as a consequence of Lemmas 3.4 & 3.5 one obtains Hamming and Lee weight distributions of Skβ . Theorem 3.6 The Hamming and Lee weight distributions of Skβ are: 1. AH (0) = 1, AH (22k−2 ) = 2k − 1, AH (2k−3 [3(2k − 1) + 1]) = 2k (2k − 1) and 2. AL (0) = 1, AL (22k−1 ) = 2k − 1, AL (2k−1(2k − 1)) = 2k (2k − 1). Proof: Similar to the proof of Theorem 3.3. Remark 3.6 (i) Skβ is of type β. (ii) The correlation of each nonzero codeword of Skβ with components 0’s or 2’s only is −2k−1 . (iii) The swe of Skβ is given as k−1 swe(x, y, z) = xn(k) + (2k − 1)x2·n(k−1) z 4 k−1 + 2 · n(k)xn(k−1) y 4 z n(k−1) , where n(k) = 2k−1(2k − 1). (iv) The minimum Euclidean weight of Skβ is 2k (3 · 2k−2 − 1). 3.3 Gray Image Families Let C be an [n, k, dH , dL] linear code over Z4 . Then φ(C), the image of C under the Gray map φ, is a binary code having 2k codewords of length 2n, and minimum Hamming 32 distance dL. However φ(C) need not be linear. Let B be the matrix (given in (2.5) for p = 2, s = 2) whose rows form a 2-basis for C and let φ(B) be the matrix obtained from B by applying gray map to each entry of B. The code Cφ generated by φ(B) is a [2n, k, ≥ ⌈ d2L ⌉] binary linear code. Note that φ(C) and Cφ have same number of codewords but in general they are not equal. The following proposition shows that both φ(S¯kα) and φ(Skβ ) are not linear. Proposition 3.2 φ(S¯kα ) and φ(Skβ ) are nonlinear for all k. Proof: Let R1 , R2 , . . . Rk be the rows of the generator matrix Gk (Gβk ). Let c = / S¯kα (Skβ ). Rk (R1 ) and let c′ = Rk−1 (Rk ) then by (3.3) ( Remark 3.6(ii) ), 2c̄ ⋆ c̄′ ∈ Hence, by Theorem 2.3, the result follows. Remark 3.7 (i) φ(S¯kα ) is a binary nonlinear code of length 22k+1 − 2 and minimum Hamming distance 22k . It meets the Plotkin bound[79] and n < 2dH . (ii) φ(Skβ ) is a binary nonlinear code of length 2k (2k − 1) and minimum Hamming distance 2k−1 (2k − 1). This is an example of a code having n = 2dH [79]. (iii) Even though, both S¯kα and Skβ are not Z2 −linear they meet the bound given by (2.10) which is true for Z2 -linear codes. Next two results are about the binary linear codes obtained from S¯kα and Skβ . Theorem 3.7 Let C = S¯kα . Then Cφ is an [22k+1 − 2, 2k, 22k ] binary linear code consisting of two copies of binary simplex code S2k with Hamming weight distribution same as the Lee weight distribution of S¯kα . 33 Proof: By Lemma 2.5, Cφ is a binary linear code of length 22k+1 − 2 and dimension 2k. Let Gk be a generator matrix of Skα in 2-basis form. Then  Gk =   0...0 1...1 2...2 3...3           0...0 2...2 0...0 2...2 Gk−1 Gk−1 Gk−1 Gk−1 2Gk−1 2Gk−1 2Gk−1 2Gk−1 Upto the suitable rearrangement of rows, φ(Gk ) is given by     .        0000 . . . 00 0101 . . . 01 1111 . . . 11 1010 . . . 10      0000 . . . 00 1111 . . . 11 0000 . . . 00 1111 . . . 11  . φ(Gk ) =     φ(Gk−1) φ(Gk−1 ) φ(Gk−1 ) φ(Gk−1)  The proof now follows by induction on k. Statement trivially holds for k = 2. Assume i h that φ(Gk−1 ) yields a 22k−1 , 2(k − 1), 22k−2 binary code in which every nonzero codeword is of weight 22k−2 . Then the possible nonzero weight from the lower portion of the above matrix φ(Gk ) will be 4 · 22k−2 = 22k . From the structure of the first two rows of φ(Gk ) it is easy to verify that any linear combination of these rows with other rows has weight 22k . Puncturing the first two columns and rearranging the columns yields the code having two copies of S2k . Theorem 3.8 Let C = Skβ . Then Cφ is the binary MacDonald code M2k,k : [22k − 2k , 2k, 22k−1 − 2k−1 ] with Hamming weight distribution same as the Lee weight distribution of Skβ . Proof: We again use induction on k. For k = 2 the result can be easily verified. If Gkβ is a generator matrix of Skβ in 2-basis form then upto rearrangement of rows    0101 · · · 01 0000 · · · 00 1111 · · · 11      φ(Gkβ ) =  1111 · · · 11 0000 · · · 00 0000 · · · 00  ,    φ(Gk−1) β φ(Gk−1 ) β φ(Gk−1 )  34 α where Gk−1 is the generator matrix of Sk−1 in 2-basis form. Assume that the result i h β β holds for Sk−1 i.e., φ(Gk−1 ) yields a 22k−2 − 2k−1 , 2k − 2, 22k−3 − 2k−2 binary code with possible nonzero weights either 22k−3 or 2k−2 (2k−1 − 1). By Theorem 3.7, φ(Gk−1 ) h i is a 22k−1 , 2(k − 1), 22k−2 binary code in which every nonzero codeword is of weight 22k−2 . Thus possible nonzero weights from lower portion of the above matrix will be   either 2(22k−3 ) + 22(k−1) or 2 22k−3 − 2k−2 + 22k−2, i.e., either 22k−1 or 2k−1 (2k − 1). Now the proof easily follows (due to structure of first two rows of the above matrix) by showing that the resulting weight of any linear combination of first two rows with lower portion of the matrix, does not changes. The weight hierarchy of Skα and Skβ are given by the following two theorems. Theorem 3.9 Skα satisfies the chain condition and its weight hierarchy is given by dr (Skα ) = r X i=1 Proof: 22k−i = 22k − 22k−r ; 1 ≤ r ≤ 2k. By Remark 3.3(1), Any r-dimensional subcode of Skα is of constant Lee weight. Hence by Remark 2.3 (for s = 2), dr (Skα ) = 1 r (2 − 1)22k = 22k − 22k−r . r 2 Let D1 =< 2R1 >, D2 =< 2R1 , 2R2 >, D3 =< R1 , 2R1 , 2R2 >, D4 =< R1 , 2R1 , R2 , 2R2 >, . . . , D2k =< R1 , 2R1 , . . . , Rk , 2Rk > . It is easy to verify that D1 ⊆ D2 ⊆ · · · ⊆ D2k , and wS (Dr ) = dr (Skα ) for 1 ≤ r ≤ 2k. Theorem 3.10 Skβ satisfies the chain condition and its weight hierarchy is given by r dr (Skβ ) = n(k) − 2k−r−1 (2k − 2⌈ 2 ⌉ ) 1 ≤ r ≤ 2k, where n(k) = 2k−1(2k − 1). 35 Proof: The proof follows by induction on k. Clearly result holds for k = 2. Assume β that the result holds for Sk−1 . Hence if 1 ≤ r ≤ 2k−2, then there exists an r-dimensional r β subcode of Sk−1 with minimum support size as n(k − 1) − 2k−r−2(2k−1 − 2⌈ 2 ⌉ ). By Remark 3.4, β α dr (Skβ ) = 2dr (Sk−1 ) + dr (Sk−1 ). (3.6) α But all r-dimensional subcodes of Sk−1 have constant support size (22k−2 − 22k−2−r ). Thus simplifying (3.6) yields the result. The case r = 2k − 1 & 2k are trivial. Let D1 =< 2R1 >, D2 =< R1 , 2R1 >, D3 =< R1 , 2R1 , 2R2 >, D4 =< R1 , 2R1 , R2 , 2R2 > , . . . , D2k =< R1 , 2R1 , . . . , Rk , 2Rk > . It is easy to see that D1 ⊆ D2 ⊆ · · · ⊆ D2k , is the required chain of subcodes. The dual code of Skα is a code of length 22k and 2-dimension 22k+1 − 2k, whereas the dual code of Skβ is a code of length 2k−1 (2k − 1) and 2-dimension 22k − 2k − 2k. The Hamming and Lee weight distributions of these dual codes can be obtained with the help of Theorems 3.3 and 3.6 and the MacWilliams Identities[92]. Similarly, the weight hierarchies of duals can be obtained from Theorems 2.6, 3.9 & 3.10. For example, we have Proposition 3.3 If 1 ≤ r ≤ 22k+1 − 2k − 2. Then {dr (Skα ⊥ )} = {1, 1, 2, 2, . . . , n, n}\{2i : 0 ≤ i < 2k}. In[108], Sun and Leib have considered the dual of Skβ (k ≥ 3) in a different context. They have used combinatorial arguments to obtain a code of length n = 2r−1 (2r − 1), q redundancy r and minimum squared noncoherent weight N +1− (N − 2)2 + 9, where N = n − 1. They have further punctured these codes to get some good codes in the sense of having larger coding gains over noncoherent detection. Chapter 4 Z2s -Simplex Codes and their Generalized Gray Images The purpose of computing is insight, not numbers. . . . Richard W. Hamming (1915-1998) 4.1 Introduction In section 2.4.2 we have studied some basic properties of generalized gray map φG introduced by Carlet. In[19] he has given two other generalizations of gray map φ. There may be several other generalizations. Thus the study of gray map on linear codes over Z4 , as seen in section 3.3 of the last chapter can be extended to codes over Z2s in several ways. In this chapter we restrict ourselves to the map φG . Simplex codes of type α and β over Z2s are studied in section 4.2. Section 4.2.1 deals with the properties of generalized gray images of Skα and Skβ . 36 37 4.2 Z2s -Simplex Codes of Type α and β Let Gk be a k × 2sk matrix over Z2s defined inductively by   Gk =   G1 = [0123 · · · 2s − 1], s s s  00 · · · 0 11 · · · 1 22 · · · 2 · · · (2 − 1)(2 − 1) · · · (2 − 1)   Gk−1 Gk−1 Gk−1 ··· ;k Gk−1 ≥ 2. Clearly, the code Skα generated by Gk over Z2s has length 2sk and 2-dimension sk. The following observations are straightforward generalizations of Remarks 3.1 and 3.2 of the previous chapter. α Remark 4.1 If Ak−1 denotes an array of codewords in Sk−1 and if i = (iii...i) then an array of all codewords of Skα is given by                Ak−1 Ak−1 Ak−1 ··· Ak−1 1 + Ak−1 2 + Ak−1 · · · (2s − 1) + Ak−1 Ak−1 .. . 2 + Ak−1 .. . 22 + Ak−1 .. . · · · (2s − 2) + Ak−1 .. .. . . Ak−1 (2s − 1) + Ak−1 (2s − 2) + Ak−1 · · · Ak−1 1 + Ak−1         .       Remark 4.2 If R1 , R2 , ..., Rk denote the rows of the matrix Gk then wH (Ri ) = 2sk − 2s(k−1) , wH (2s−1 Ri ) = 2sk−1 , wL(Ri ) = 2s(k+1)−2 , and wGL (Ri ) = 2s(k+1)−2 . For each m, 0 ≤ m ≤ s, let Sm = {0, 1 · 2s−m , 2 · 2s−m , · · · , (2m − 1) · 2s−m }. Note that Ss−1 = Z, the set of all zero divisors of Z2s and Ss = Z2s . A codeword c = (c1 , . . . , cn ) ∈ Skα is said to be of type m if all of its components belong to the set Sm . It may be observed that each element of Z2s occurs equally often in every row of Gk . In fact we have the following Lemma. Lemma 4.1 Let c ∈ Skα be a type m codeword. Then all the components of c will occur equally often 2sk−m times. 38 Proof: y1 = y3 =   y2s =  α Any x ∈ Sk−1 gives rise to following 2s codewords of Skα x x x ··· x  , y2 = x 2 + x 22 + x · · · (2s − 2) + x s   s x 1 + x 2 + x · · · (2 − 1) + x  , , . . . , and s x (2 − 1) + x (2 − 2) + x · · · 1 + x  . Hence the proof follows easily by induction on k and the Remark 4.1. To determine weight distribution of Skα one needs to determine the number of codewords of type m in Skα for 1 ≤ m ≤ s. Let Cm be the matrix defined by  Cm =                                  2 s−m .. .  R1  2s−1 R1 2s−mR2 .. . 2s−1 R2 .. . 2s−m Rk .. . 2s−1Rk                ,                 where Ri is the ith row of the matrix Gk . The subcodes D (m) of C generated by the 2-linear combinations of the rows of Cm will have 2mk codewords. Note that the codewords generated by the matrix C1 have components either 0 or 2s−1 and Cs yields the whole code Skα . Thus, for all m, 1 ≤ m ≤ s, a codeword of type m will occur 2mk − 2(m−1)k times in Skα . This proves the following Lemma. Lemma 4.2 Let 0 < m ≤ s. Then the number of codewords of type m in Skα is 2(m−1)k (2k − 1). 39 Theorem 4.3 The Hamming, Lee and the generalized Lee weight distributions of Skα are: 1. AH (0) = 1, AH (2sk−m(2m − 1)) = 2(m−1)k (2k − 1) for 1 ≤ m ≤ s, 2. AL (0) = 1, AL (2s(k+1)−2 ) = 2sk − 1 and 3. AGL (0) = 1, AGL (2s(k+1)−2 ) = 2sk − 1. Let c ∈ Skα be a codeword of type m(6= 0). Then by Lemma 4.1, wH (c) = Proof: 2sk − 2sk−m and hence by Lemma 4.2, AH (2sk−m(2m − 1)) = 2(m−1)k (2k − 1). For m = 0, AH (0) = 1. Also, by Lemma 4.1 wL(c) = 2sk−m P m (2 −1) t=0  wL(t · 2s−m) = 2s(k+1)−2 which is independent of m. Thus all type m(6= 0) codewords will have same Lee weight. Similar argument holds for generalized Lee weight. Remark 4.3 (i) Skα is an equidistant code with respect to Lee and generalized Lee distances. (ii) Skα is of type α. (iii) For s = 1, Skα reduces to extended binary simplex code Sˆk . (iv) In[97], Satyanarayna also visited the codes Skα and obtained the second part of the above theorem. Let Gβk be the k × 2(s−1)(k−1) (2k − 1) matrix defined inductively by   and for k > 2,   Gβk =   Gβ2 =   111 · · · 1 s  s 0 2 4 6 · · · (2 − 2)  0123 · · · (2 − 1) 1 1 1 1 · · · 1 ,  s s  11 · · · 1 00 · · · 0 22 · · · 2 44 · · · 4 · · · (2 − 2) · · · (2 − 2)  Gk−1 Gβk−1 Gβk−1 Gβk−1 ··· Gβk−1 ,  40 α where Gk−1 is the generator matrix of Sk−1 . By induction, it is easy to verify that no two columns of Gβk are multiple of each other. Let Skβ be the linear code over Z2s generated by Gβk . Note that Skβ is a [2(s−1)(k−1) (2k − 1), sk] code. Now the Remarks 3.4 & 3.5 of previous chapter generalize to the following. β α Remark 4.4 If Ak−1 (Bk−1) denotes an array of codewords in Sk−1 (Sk−1 ) and if i = (i, i, . . . , i) then an array of all codewords of Skβ is given by               Ak−1 Bk−1 Bk−1 Bk−1 ··· Bk−1 1 + Ak−1 Bk−1 2 + Bk−1 22 + Bk−1 ··· (2s − 2) + Bk−1 2 + Ak−1 .. . Bk−1 .. . 22 + Bk−1 .. . 23 + Bk−1 .. . ··· .. . (2s − 22 ) + Bk−1 .. . (2s − 1) + Ak−1 Bk−1 (2s − 2) + Bk−1 (2s − 22 ) + Bk−1 · · · 2 + Bk−1        .       Remark 4.5 Each row of Gβk has Hamming weight 2(s−1)(k−1)−s [(2k − 1)(2s − 1) + 1], and Generalized Lee weight 2sk−k−1(2k − 1). The Lee weight of the first row will be 2s(k−1) + 2sk−2 − 2sk−k−1. Let U, Z be the set of units and zero divisors of Z2s , respectively. Proposition 3.1 generalizes to the following. Proposition 4.1 Let 1 ≤ j ≤ k and let Rj be the j th row of Gβk . Then P i∈U ωi = 2s(k−1) , and each zero divisor in Z2s occurs 2(s−1)(k−2) (2k−1 − 1) times in Rj . Proof: It follows by induction and is similar to the proof of proposition 3.1 The next Proposition is similar to Lemma 4.2 of Skα . Proposition 4.2 Let c ∈ Skβ . If one of the coordinates of c is a unit then 2s(k−1) , and each zero divisor in Z2s occurs 2(s−1)(k−2) (2k−1 − 1) times in c. P i∈U ωi = 41 β α By Remark 4.4, there exist y1 ∈ Sk−1 and y2 ∈ Sk−1 such that c can have Proof: any of the following 2s forms: c= c= c=    y1 y2 y2 · · · y2  , c=  2 + y1 y2 22 + y2 · · · (2s − 22 ) + y2 s s 1 + y1 y2 2 + y2 · · · (2 − 2) + y2  s , . . . , or (2 − 1) + y1 y2 (2 − 2) + y2 · · · 2 + y2  . The proof now follows on the lines of the proof of Lemma 3.4. Let C be a linear code over Z2s . We can define the reduction code C (1) and the torsion code C (2) of C as follows. Let D = {c ∈ C|ci = 0 or 2s−1 for all i}. Then 1 C (2) = { 2s−1 c|c ∈ D} is the torsion code of C. The binary code C (1) = C (mod 2) is called the reduction code of C. If C is a free module then C (2) = C (1) . Hence the   reduction and torsion codes of Skα Skβ are equal. Next proposition determines these binary codes. Proposition 4.3 The torsion code of Skα    Skβ is equivalent to 2(s−1)k copies of the  extended binary simplex code 2(s−1)(k−1) copies of the binary simplex code . Proof: The proof is by induction on k and follows on the lines of the proof of Lemmas 3.2 and 3.5. Theorem 4.4 The Hamming and Generalized Lee weight distributions of Skβ are 1. AH (0) = 1, AH (2(s−1)(k−1) [2k−m {2m − 1} + {21−m − 1}]) = 2(m−1)k (2k − 1), for each m; 1 ≤ m ≤ s, and 2. AGL (0) = 1, AGL (2sk−1) = 2k − 1, AGL (2sk−k−1(2k − 1)) = 2k (2(s−1)k − 1). Proof: By induction on k. By Theorem 4.3 and Remark 4.4 it is easy to see that the possible nonzero Hamming (Generalized Lee) weights of Skβ are n   2(s−1)(k−1) 2k−m (2m − 1) + (21−m − 1) : 1 ≤ m ≤ s o n o 2sk−1 , 2sk−k−1(2k − 1) .  , 42 By Lemma 4.2, Hamming weight of type m will occur 2(m−1)k (2k − 1) times. Moreover generalized lee weight 2sk−1 will occur 2k − 1 times. Thus the other weight will occur 2sk − 2k times. Corollary 4.5 (i) Skβ is of type β. (ii) For s = 1, Skβ reduces to binary Simplex Code Sk . 4.2.1 Gray Image Families Let C be an [n, k, dH , dGL] linear code over Z2s and let φG be the Generalized Gray map defined in section 2.4.2. Then φG (C) is a binary code having 2k codewords of length 2s−1 n, and minimum Hamming distance dGL. However φG (C) need not be linear. Let B be the matrix given in (2.5) for p = 2. The rows of B form a 2-basis for C and let φG (B) be the matrix obtained from B by applying the generalized gray map φG to each GL ⌉] binary linear code. entry of B. The code Cφ generated by φG (B) is a [2s−1 n, k, ≥ ⌈ 2ds−1 Note that φG (C) and Cφ have the same number of codewords but they are not equal in general. Recall that S¯kα is the punctured code Skα . Remark 4.6 (i) φG (S¯kα ) is a binary code of length 2s−1 (2sk − 1) and minimum Hamming distance 2s(k+1)−2 . It meets the Plotkin bound (cf.[79]) and n < 2dH . (ii) φG (Skβ ) is a binary code of length 2k(s−1) (2k − 1) and minimum Hamming distance 2sk−k−1(2k − 1). This is an example of a code having n = 2dH (cf.[79]). The next two results are about the binary linear codes obtained from S¯kα and Skβ . Theorem 4.6 Let C = S¯kα . Then Cφ is a [2s−1 · (2sk − 1), sk, 2s(k+1)−2] binary linear code consisting of 2s−1 copies of the binary simplex code Ssk with Hamming weight dis- 43 tribution same as the Generalized Lee weight distribution of S¯kα . Proof: By Lemma 2.5, Cφ is a binary linear code of length 2s−1 · (2sk − 1) and dimension sk. Let Gk be a generator matrix of Skα in 2-basis form. Then               Gk =               0...0 1...1 2...2 3...3 ... 0...0 2...2 4...4 3 · 2...3 · 2 ... 22 . . . 3 · 22 ... .. .  (2s − 1) . . . (2s − 1)  (2s 22 ) . . . (2s − 22 ) 0...0 .. . 4...4 .. . 8...8 .. . 3· 0...0 2s−1 . . . 2s−1 0...0 2s−1 . . . 2s−1 ... 2s−1 . . . 2s−1 Gk−1 Gk−1 Gk−1 Gk−1 ... Gk−1 2Gk−1 .. . 2Gk−1 .. . 2Gk−1 .. . 2Gk−1 .. . ... .. . 2Gk−1 .. . 2s−1 Gk−1 2s−1 Gk−1 2s−1 Gk−1 2s−1 Gk−1 ... 2s−1 Gk−1 .. .  (2s − 2) . . . (2s − 2)   − .. .                        and φG (Gk ) =        φG (2s − 1) . . . φG (2s − 1) .. . 0000 . . . 00 .. . 0101 . . . 01 .. . 0011 . . . 11 .. . 0110 . . . 10 .. . ... 0000 . . . 00 1111 . . . 11 0000 . . . 00 1111 . . . 11 ... 1111 . . . 11 φG (Gk−1 ) φG (Gk−1 ) φG (Gk−1 ) φG (Gk−1 ) ... φG (Gk−1 ) .. .     .   The proof is by induction on k and proceeds on the lines of the proof of Theorem 3.7. For k = 2, result follows trivially. Assume that result holds for k − 1, i.e, φG (Gk−1 ) yields a binary code in which every nonzero codeword is of weight 2sk−2. Then by induction hypothesis the possible nonzero weight from the lower portion of the matrix φG (Gk ) will be 2s · 2sk−2 = 2s(k+1)−2 . From the structure of the first s rows of φ(Gk ) it is easy to verify that any linear combination of these rows with other rows coming from the lower portion has weight 2sk+s−2. Puncturing the first 2s−1 columns corresponding to the first column of Gk and rearranging the rest of the columns yields the code having 2s−1 copies of Ssk . 44 Theorem 4.7 Let C = Skβ . Then Cφ is the binary MacDonald code Msk,(s−1)k : [2sk − 2(s−1)k , sk, 2sk−1 − 2(s−1)k−1 ] with Hamming weight distribution same as the Generalized Lee weight distribution of Skβ . Proof: By induction on k and is similar to the proof of Theorem 3.8. Let Gkβ be a generator matrix of Skβ in 2-basis form then φG (Gkβ ) =        φG (2s − 2) · · · φG (2s − 2) .. . 0101 · · · 01 .. . 0000 · · · 00 .. . 0011 · · · 11 .. . 0000 · · · 1111 .. . ··· 1111 · · · 11 0000 · · · 00 0000 · · · 00 0000 · · · 0000 ··· 0000 · · · 0000 φG (Gk−1 ) β φG (Gk−1 ) β φG (Gk−1 ) β φG (Gk−1 ) ··· β φG (Gk−1 ) .. .     ,   α where Gk−1 is the generator matrix of Sk−1 in 2-basis form. It is easy to verify β β that result holds for k = 2. Assume that result holds for Sk−1 . Then φG (Gk−1 ) yields a binary code with possible nonzero weights either 2sk−s−1 or 2sk−s−k (2k−1 − 1). By Theorem 4.6, φG (Gk−1 ) is a binary code in which every nonzero codeword is of weight 2sk−2. Thus possible nonzero weights from lower portion of the above matrix will be   either 2s−1(2sk−s−1) + 2sk−2 or 2s−1 2sk−s−1 − 2sk−s−k + 2sk−2 . Now the proof follows (easily from the structure of first s rows of the above matrix) by showing that the resulting weight, of any linear combination of first s rows with lower portion of the matrix, does not changes. The weight hierarchy of Skα and Skβ are given by the following two theorems. Theorem 4.8 Skα satisfies the chain condition and its weight hierarchy is given by dr (Skα ) = r X i=1 2sk−i = 2sk − 2sk−r ; 1 ≤ r ≤ sk. 45 Proof: By Remark 4.3, any r-dimensional subcode of Skα is of constant Generalized Lee weight. Hence by definition (see Remark 2.3) dr (Skα ) = 1 2r+s−2 X c(6=0)∈Dr wGL(c) = 2−r−s+2 · 2sk+s−2 · X c(6=0)∈Dr 1 = 2sk−r (2r − 1). Let D1 =< 2s−1R1 >, D2 =< 2s−1 R1 , 2s−1 R2 >, . . . , Dk =< 2s−1 R1 , . . . , 2s−1Rk > , Dk+1 =< 2s−2 R1 , 2s−1 R1 , 2s−1 R2 , . . . , 2s−1Rk >, . . . , Dsk =< R1 , 2R1 , . . . , 2s−1R1 , . . . , Rk , 2Rk , . . . , 2s−1Rk > . It is easy to verify that D1 ⊆ D2 ⊆ · · · ⊆ Dsk , and wS (Dr ) = dr (Skα ) for 1 ≤ r ≤ sk. Theorem 4.9 Weight hierarchy of Skβ is given by dr (Skβ ) = n(k) − 2(s−1)(k−1) (2k−r − 2i−r ), where 1 ≤ i ≤ k, (i − 1)s < r ≤ is and n(k) = 2(s−1)(k−1) (2k − 1). Moreover Skβ satisfies the chain condition. Proof: The proof follows by induction on k. Clearly the result holds for k = 2. β Assume that the result holds for Sk−1 . Hence, if 1 ≤ i ≤ k − 1, then there exβ ists an r-dimensional subcode of Sk−1 with minimum support size as n(k − 1) − 2(s−1)(k−2) (2k−1−r − 2i−r ). By Remark 4.4, β α dr (Skβ ) = 2s−1 dr (Sk−1 ) + dr (Sk−1 ). (4.1) α But all r-dimensional subcodes of Sk−1 have constant support size (2sk−s − 2sk−s−r ). Thus simplifying (4.1) yields the result. The case i = k is trivial. Let D1 =< 2s−1R1 > , D2 =< 2s−2 R1 , 2s−1R1 >, . . . , Ds =< R1 , 2R1 , 22 R2 , . . . , 2s−1 R1 >, Ds+1 =< R1 , 2R1 , 22 R2 , . . . , 2s−1 R1 , 2s−1 R2 >, . . . , Ds+s =< R1 , 2R1 , 22 R2 , . . . , 2s−1 R1 , R2 , 2R2 , . . . , 2s−1R2 >, . . . , Dsk =< R1 , 2R1 , 22 R2 , . . . , 2s−1 R1 , R2 , 2R2 , . . . , 2s−1R2 , . . . , Rk , 2Rk , 22 Rk , . . . , 2s−1 Rk > . It is easy to see that D1 ⊆ D2 ⊆ · · · ⊆ Dsk 46 is the required chain of subcodes. The dual code of Skα is a code of length 2sk and 2-dimension s(2sk − k), whereas the dual code of Skβ is a code of length 2(s−1)(k−1) (2k − 1) and 2-dimension   s 2(s−1)(k−1) (2k − 1) − k . The weight hierarchies of duals can be obtained from Theorem 2.6, 4.8 & 4.9. Chapter 5 Codes Satisfying the Chain Condition As the births of living creatures, at first, are ill-shapen: so are all Innovations, which are the births of time. . . . Francis Bacon (1561-1626) The concept of chain condition for linear codes over GF (q) was found useful in expressing weight hierarchies of a product code in terms of the weight hierarchies of its component codes. In Chapter 2, we extended the concept of chain condition to codes over Zps . In this chapter, it is shown that various known self-dual codes over Z4 satisfy the chain condition. Construction and properties of first order Reed Müller code over Z2s are given in section 2. 5.1 Self-Dual and Self Orthogonal Codes over Z4 Self dual and self orthogonal codes over Z4 were recently studied by several researchers like Bonnecaze, Conway, Harada, Pless, Quian, Rains and Sloane etc. (see[26],[11],[12], [33],[50],[39],[54],[86],[64],[87],[93],[92],[88] etc.). The binary image under the gray map φ of a self dual code over Z4 is formally self-dual and distance-invariant[53]. Let C be 47 48 a linear code over Z4 . A codeword c ∈ C is said to be a tetrad if it has exactly four coordinates congruent to 1 or 3 (mod 4) and the rest congruent to 0 (mod 4). Let m ≥ 2 be a positive integer. Let D2m be the [2m, 2m − 2, 4, 4] type β code generated by (m − 1) × 2m matrix             1 1 1 3 0 0 ··· 0 0 0 0 0   0 .. . 0 .. . 1 .. . 1 .. . 1 .. . 3 ··· 0 0 .. . . .. .. . . . . 0 .. . 0 .. . 0 .. . 0 0 0 0 0 0 ··· 0 1 1 1 3    ,     (5.1) 0 and let D2m be the [2m, 2m − 1] code generated by D2m and the tetrads 1300 . . . 0011. 0 Equivalently D2m is generated by the matrix (see[26])                 1 1 1 3 0 0 ··· 0 0 0 0 0  0 .. . 0 .. . 1 .. . 1 .. . 1 .. . 3 ··· 0 0 .. . . .. .. . . . . 0 .. . 0 .. . 0 .. . 0 0 0 0 0 0 ··· 0 1 1 1 3 2 0 2 0 2 0 ··· 0 2 0 2 0       .       (5.2) + ⊕ Let D2m be the [2m, 2m − 1] code generated by D2m and 00 . . . 0022 and let D2m be + ⊕ 0 the code generated by D2m and D2m . Note that D2m is a [2m, 2m] self-dual code([26]). Let E7 be the [7, 6, 4, 4] type β code generated by the matrix         1 0 0 3 1 1 0    1 0 1 0 0 3 1   1 1 0 1 0 0 3 (5.3)  and let E7+ be the [7, 7, 4, 4] type β code generated by E7 and 2222222. It was observed in[26] that E7+ is a self-dual code and the reduction code of both E7 & E7+ is the 49 Hamming code of length 7. They also show that the code E8 generated by the matrix             1 0 0 0 0 1 1 1    0 1 0 0 3 0 1 3     (5.4) 0 0 1 0 3 3 0 1   0 0 0 1 3 1 3 0  is a [8, 8, 4, 4] self dual code. Note that it is a type β code. In[26] Conway and Sloane has shown that any self orthogonal code over Z4 gen+ ⊕ 0 erated by ‘tetrads’ is equivalent to a direct sum of codes D2m , D2m , D2m , D2m (m = 1, 2, . . . ), E7 , E7+ , E8 . The following Theorem shows that D2m , E7 , E7+ and E8 satisfy the chain condition. Note that the smallest self-dual code A1 = {0, 2} trivially satisfies the chain condition. Theorem 5.1 D2m , E7 , E7+ and E8 satisfies chain conditions. Proof: It is easy to verify that the weight hierarchy of E7 is {4, 4, 6, 6, 7, 7}. Con- sider the codewords x1 = (1101003), x2 = (2111030) and x3 = (3023132) of E7 . Let D1 =< 2x1 >, D2 =< x1 , 2x1 >, D3 =< x1 , 2x1 , 2x2 >, D4 =< x1 , 2x1 , x2 , 2x2 >, D5 =< x1 , 2x1 , x2 , 2x2 , 2x3 >, and D6 =< x1 , 2x1 , x2 , 2x2 , x3 , 2x3 > . It is easy to verify that D1 ⊆ D2 . . . ⊆ D6 is the required chain of subcodes. Therefore for E7+ the required chain of subcodes can be taken as D1 ⊆ D2 . . . ⊆ D6 ⊆ D7 where Di for 1 ≤ i ≤ 6 are the subcodes defined for E7 and D7 =< x1 , 2x1 , x2 , 2x2 , x3 , 2x3 , 2222222 > . Hence E7+ satisfies the chain condition and its weight hierarchy is {4, 4, 6, 6, 7, 7, 8}. Clearly, the weight hierarchy of E8 is {4, 4, 6, 6, 7, 7, 8, 8}. If Ri (1 ≤ i ≤ 4) denote the first four rows of the matrix given in (5.4) and if D1 =< 2R4 >, D2 =< R4 , 2R4 >, D3 =< 2R3 , R4 , 2R4 >, D4 =< R3 , 2R3 , R4 , 2R4 >, . . . , D8 =< R1 , 2R1 , . . . , R3 , 2R3 , R4 , 2R4 > then D1 ⊆ D2 . . . ⊆ D8 and ws (Dr ) = dr (E8 ). 50 It is easy to see that the code D2m has weight hierarchy {4, 4, 6, 6, 8, 8, . . . , 2m − 2, 2m − 2, 2m, 2m}. Let R1 , . . . Rm−1 be the first m − 1 rows of the matrix given in (5.1) and let D1 =< 2R1 >, D2 =< R1 , 2R1 >, . . . , D2m−3 =< R1 , 2R1 , . . . , Rm−2 , 2Rm−2 , 2Rm−1 >, D2m−2 =< R1 , 2R1 , . . . , Rm−2 , 2Rm−2 , Rm−1 , 2Rm−1 > . Then D1 ⊆ D2 . . . ⊆ D2m−2 and ws (Dr ) = dr (D2m ), 1 ≤ r ≤ 2m − 2. There is another self-dual code L8 of length 8 defined in[26]. L8 is [8, 8, 2, 4] type α code generated by the matrix                 00 11 02 13    00 02 13 11     . 11 02 00 13   02 02 02 02 00 00 00 22 (5.5)      Proposition 5.1 L8 satisfies the chain condition. Proof: It is easy to see that the weight hierarchy of L8 is {2, 4, 5, 6, 7, 7, 8, 8}. Let x1 = (00000022), x2 = (11020013), x3 = (13000211), x4 = (02001131) and x5 = (13021122) be the five codewords of L8 . Let D1 =< x1 >, D2 =< 2x2 , x1 >, D3 =< x2 , 2x2 , x1 >, D4 =< x3 , x2 , 2x2 , x1 >, D5 =< 2x4 , x3 , x2 , 2x2 , x1 >, D6 =< x4 , 2x4 , x3 , x2 , 2x2 , x1 >, D7 =< 2x5 , x4 , 2x4 , x3 , x2 , 2x2 , x1 >, and D8 =< x5 , 2x5 , x4 , 2x4 , x3 , x2 , 2x2 , x1 > . Then D1 ⊆ D2 . . . ⊆ D8 and ws (Dr ) = dr (L8 ), 1 ≤ r ≤ 8. Let m ≥ 1. Let K4m be the [4m, 4m, 2, 4] type α code generated by the (4m−1)×4m 51 matrix                ′  1 1 1 ··· 1 1  0 2 0 ··· 0 2 0 .. . 0 .. . 2 ··· 0 2 .. . . .. .. . . . . 0 0 0 ··· 2 2              (5.6) and let K8 be the [8, 8, 2, 4] type α code generated by the matrix                     1 1 1 1 0 0 0 2    0 0 0 2 1 1 1 1     0 2 0 2 0 0 0 0   .  (5.7) 0 0 2 2 0 0 0 0     0 0 0 0 0 2 0 2    0 0 0 0 0 0 2 2 Both of these self dual codes can also be obtained from a labeled graph[26]. The code K4m was first constructed by Klemm[71]. ′ Theorem 5.2 Klemm Code K4m (m ≥ 1) and the code K8 satisfy the chain condition and weight hierarchy of K4m is given by dr (K4m ) = Proof:      r + 1, 1 ≤ r ≤ 4m − 2, 4m, r = 4m − 1 & 4m. (5.8) It is easy to see that the weight hierarchy of the Klemm code is given by (5.8). Let Ri (1 ≤ i ≤ 4m − 1) be the first 4m − 1 rows of (5.6). For 1 ≤ r ≤ 4m − 2, let Dr =< R4m−i : 1 ≤ i ≤ r > also let D4m−1 =< 2R1 , R2 , R3 , · · · , R4m−2 , R4m−1 > , D4m =< R1 , 2R1 , R2 , R3 , · · · , R4m−2 , R4m−1 > . Then D1 ⊆ D2 . . . ⊆ D4m and ws (Dr ) = dr (K4m ), 1 ≤ r ≤ 4m. ′ For the code K8 it can be easily checked that its weight hierarchy is {2, 3, 4, 5, 6, 7, 8, 8}. Let x1 = (00000022), x2 = (00000202), x3 = (00021111), x4 = (00201111), 52 ′ x5 = (02001111), x6 = (20001111) and x7 = (20001133) be the seven codewords of K8 . Then D1 =< x1 >, D2 =< x2 , x1 >, D3 =< 2x3 , x2 , x1 >, D4 =< x3 , 2x3 , x2 , x1 >, D5 =< x4 , x3 , 2x3 , x2 , x1 >, D6 =< x5 , x4 , x3 , 2x3 , x2 , x1 >, D7 =< x6 , x5 , x4 , x3 , 2x3 , x2 , x1 > and D8 =< x7 , x6 , x5 , x4 , x3 , 2x3 , x2 , x1 > form the required chain of subcodes. Some of the self-dual Z4 -codes have the property that all Euclidean weights are multiple of 8 and they contain the all-one vector. These codes are called Type-II codes over Z4 (see[54],[26]). The key motivation to study these codes is that one can associate a Type-II even unimodular lattice via the construction A (mod 4)[92],[16]. Several such Type-II lattices have been obtained in such a fashion. Quadratic residue codes over Z4 is a well known family of Type-II codes. These codes are obtained by Hensel uplifting of the binary quadratic residue codes[88],[54]. If n = q + 1 and q ≡ −1 (mod 8) is a prime power then Pless and Quian have shown that an extended quadratic residue code QRn of length n is a Type II code[88]. These have been widely studied by Pless et al for n = 8, 24, 32, 48 etc.[88],[87]. QR8 is the well known Octacode generated by the matrix             3 3 2 3 1 0 0 0  3 0 3 2 3 1 0 0 3 0 0 3 2 3 1 0 3 0 0 0 3 2 3 1     .     (5.9) It is an [8, 8, 4, 6] code of type β. The code QR24 is the well known lifted Golay Code. It is a [24, 24, 8, 12] code of type β generated by the matrix 53                           3 3 1 0 0 2 3 3 3 0 3 2 1 0 0 0 0 0 0 0 0 0 0 0 3 0 3 1 0 0 2 3 3 3 0 3 2 1 0 0 0 0 0 0 0 0 0 0 3 0 0 3 1 0 0 2 3 3 3 0 3 2 1 0 0 0 0 0 0 0 0 0 3 0 0 0 3 1 0 0 2 3 3 3 0 3 2 1 0 0 0 0 0 0 0 0 3 0 0 0 0 3 1 0 0 2 3 3 3 0 3 2 1 0 0 0 0 0 0 0 3 0 0 0 0 0 3 1 0 0 2 3 3 3 0 3 2 1 0 0 0 0 0 0 3 0 0 0 0 0 0 3 1 0 0 2 3 3 3 0 3 2 1 0 0 0 0 0 3 0 0 0 0 0 0 0 3 1 0 0 2 3 3 3 0 3 2 1 0 0 0 0 3 0 0 0 0 0 0 0 0 3 1 0 0 2 3 3 3 0 3 2 1 0 0 0 3 0 0 0 0 0 0 0 0 0 3 1 0 0 2 3 3 3 0 3 2 1 0 0 3 0 0 0 0 0 0 0 0 0 0 3 1 0 0 2 3 3 3 0 3 2 1 0 3 0 0 0 0 0 0 0 0 0 0 0 3 1 0 0 2 3 3 3 0 3 2 1              .             (5.10) The following theorem shows that these codes satisfy the chain condition. Theorem 5.3 Extended quadratic residue codes QR8 and QR24 over Z4 satisfy the chain condition. Proof: It is easy to see that the weight hierarchy of the Octacode is {4, 5, 6, 6, 7, 7, 8, 8} (see also[1]). Let Ri , 1 ≤ i ≤ 4, be the last four rows of the matrix (5.9). Let D1 =< 2R1 >, D2 =< R1 , 2R1 >, D3 =< 2R2 , R1 , 2R1 >, D4 =< R2 , 2R2 , R1 , 2R1 >, D5 =< 2R3 , R2 , 2R2 , R1 , 2R1 >, D6 =< R3 , 2R3 , R2 , 2R2 , R1 , 2R1 >, D7 =< 2R4 , R3 , 2R3 , R2 , 2R2 , R1 , 2R1 >, and D8 =< R4 , 2R4 , R3 , 2R3 , R2 , 2R2 , R1 , 2R1 > . Then it is easy to verify that D1 ⊆ D2 . . . ⊆ D8 and ws (Dr ) = dr (QR8 ), 1 ≤ r ≤ 8. It can be seen easily that {8, 10, 12, 13, 14, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24} is the weight hierarchy of QR24 . Let Ri ; 1 ≤ i ≤ 12 be the first 12 rows of (5.10). Let D1 =< 2R12 >, D2 =< R12 , 2R12 >, D3 =< R12 , 2R12 , 2R11 >, . . . , D24 =< R1 , 2R1 , . . . , R12 , 2R12 > . Then D1 ⊆ D2 . . . ⊆ D24 and ws (Dr ) = dr (QR24 ), 1 ≤ r ≤ 24. 54 5.2 First order Reed Müller Code over Z2s In[53], Hammons et al have constructed a linear code over Z4 (called Quaternary first order Reed Müller code) whose gray image is the binary first order Reed Müller code. In this section we construct a linear code over Z2s whose image under the generalized gray map φG is the binary first order Reed Müller code. Some basic properties of these are also obtained. Let 1 ≤ i ≤ m − s + 1. Let vi be a vector of length 2m−s+1 consisting of successive blocks of 0′ s and 1′ s each of size 2(m−s+1)−i and let 1 = (111 . . . 11) ∈ m−s+1 Z22 . Let G be a (m − s + 2) × 2m−s+1 matrix given by (consisting of the rows as 1 and 2s−1 vi (1 ≤ i ≤ m − s + 1))  G=           1 0 .. . 1 0 .. . ··· 1 ··· 0 . . .. . . 1 1 1 0 .. . s−1 s−1 2 2 .. . 0 2s−1 · · · 0 2s−1 ··· 1 s−1 s−1 ··· 2 .. .. . . .. . 2s−1 · · · 0 1 2 .. . 2s−1 0            (5.11) The code generated by G is called the first order Reed Müller code over Z2s , denoted R1,m−s+1 . It is a [2m−s+1 , m + 1, 2m−s , 2m−s+1 , 2m−1 ] linear code over Z2s . It’s generator matrix in 2-basis form is given by  G=                       1 1 ··· 1 1 1 1 ··· 1 1 2 .. . 2 .. . ··· .. . 2 .. . 2 .. . 2 .. . 2 .. . ··· .. . 2 .. . 2 .. . 2s−1 2s−1 · · · 2s−1 2s−1 2s−1 2s−1 · · · 2s−1 2s−1 0 .. . 0 0 .. . ··· .. . 0 .. . 0 .. . 2s−1 · · · 0 2s−1 2s−1 2s−1 · · · 2s−1 2s−1 .. .. .. .. .. . . . . . 0 2s−1 · · · 0 2s−1                        (5.12) 55 Remark 5.1 For s = 2, R1,m−s+1 reduces to Quaternary first order Reed Müller Code ZRM(1, m) defined in[53]. Proposition 5.2 Hamming and Generalized Lee weight distributions of R1,m−s+1 are: 1. AH (0) = 1, AH (2m−s ) = 2m−s+2 − 2 & AH (2m−s+1 ) = 2m+1 − 2m−s+2 + 1, 2. AGL (0) = 1, AGL (2m ) = 1 & AGL (2m−1 ) = 2m+1 − 2. Proof: Note that first s rows of the matrix G given by (5.12) have Hamming weight 2m−s+1 and remaning m − s + 1 rows are of Hamming weight 2m−s . It is easy to see that any 2-linear combination of last m − s + 1 rows also has Hamming weight 2m−s . Hence AH (2m−s ) = 2m−s+2 − 2. Since −2s−1 = 2s−1 in Z2s , every other nontrivial 2linear combination has weight 2m−s+1 . Thus AH (2m−s+1 ) = 2m+1 − 2m−s+2 + 1. Similar arguments holds for generalized Lee weight. Theorem 5.4 Weight hierarchy of R1,m−s+1 is given by  Pt−1 m−s−i   , i=0 2 dt (R1,m−s+1 ) =   2m−s+1 , 1≤t≤m−s+1 m − s + 1 < t ≤ m + 1. Moreover, R1,m−s+1 satisfies the chain condition. Proof: By Corollary 2.9, dt (R1,m−s+1 ) ≥ & t m−1 (2 − 1)2 2t+s−2 ' =  Pt−1 m−s−i   2 , i=0   2m−s+1 , 1≤t≤m−s+1 m − s + 1 < t ≤ m + 1. (5.13) Let 1 ≤ t ≤ m − s + 1. Let D be a t-dimensional subcode of R1,m−s+1 generated by any t rows chosen from last m − s + 1 rows of (5.12). Thus chosen t rows shares 2m−s+1−t common zero bit positions. Hence the support size of D is t > m − s + 1 then trivially equality holds in (5.13). Pt−1 i=0 2m−s−i. If 56 Let D1 =< 2s−1vm−s+1 >, D2 =< 2s−1 vm−s , 2s−1 vm−s+1 >, . . . , Dm−s+1 =< 2s−1 v1 , . . . , 2s−1 vm−s , 2s−1vm−s+1 >, Dm−s+2 =< 2s−1 , 2s−1 v1 , . . . , 2s−1vm−s , 2s−1 vm−s+1 >, . . . , and Dm−1 =< 1, 2, 22 , . . . , 2s−1 , 2s−1 v1 , . . . , 2s−1vm−s , 2s−1 vm−s+1 > . Then D1 ⊆ D2 . . . ⊆ Dm+1 and ws (Dr ) = dr (R1,m−s+1 ); 1 ≤ r ≤ m + 1. The map φG is nonlinear over Z2s as φG (2 + 3) 6= φG (2) + φG (3). However we have the following Lemma. Lemma 5.5 Let T = {2i : 0 ≤ i ≤ s − 1} ∪ {0}. Then φG (a + b) = φG (a) + φG (b), for all a, b ∈ T. Proof: The verification follows easily using the structure of Es given by (2.6) and the definition of φG in (2.7). Theorem 5.6 R1,m−s+1 is Z2 -linear. Proof: Let c ∈ R1,m−s+1 . Then c can be written as a 2-linear combination of the rows of the matrix G given by (5.12). Since φG maps G to a generator matrix of a binary first order Reed Müller code (with some permutation of the rows, see (2.9, 5.12 and the Lemma 5.5). Thus φG (c) belongs to the binary linear first order Reed Müller code. Hence R1,m−s+1 is Z2 -linear. Chapter 6 Norm Quadratic Residue Codes Often you have to rely on intuition. . . . Bill Gates (1955-till date) In this chapter we investigate a weakly self dual family of binary linear codes, called, NQR Codes based on the concept of quadratic residues and obtain its basic properties over binary field. It is shown that NQR Codes are Z4 -linear. 6.1 Definitions and Basic Results Let p be an odd prime. The Poincáre finite upper half plane is the set Hp := {x + √ δy x, y ∈ GF (p), y 6= 0, δ is a quadratic non residue (q.n.r.) (mod p)}. (6.1) In[110] Tiu and Wallace have used it to construct a new class of binary linear codes Cp called norm quadratic residue (NQR) codes. The length of each codeword in Cp is p(p − 1), the number of points in Hp . The points in Hp are ordered lexicographically; i.e, (x, y) < (u, v) if x < u or x = u and y < v. For each a ∈ GF (p), the set Ba = {(a, 1), (a, 2), . . . , (a, p − 1)} is called a block. Let G be the (p + 1) × p(p − 1) 57 58 matrix  G= p(p−1) where each Ri ∈ IF 2            R0  R1 .. . Rp          is defined as follows. R0 has an 1 in position (x, y) if x2 − δy 2 , the norm of (x, y) in Hp , is a quadratic residue (q.r.) mod p and 0, otherwise. R1 , R2 , . . . , Rp−1 are block wise cyclic shifts of R0 corresponding to blocks B0 , B1 , . . . , Bp−1 , and Rp is the all one vector. The set of entries in R0 corresponding to the block Ba will be denoted by ba . For example, for p = 7, the columns of G are grouped corresponding to blocks B0 , B1 , · · · , B6 , and the corresponding generator matrix of the NQR code C7 with δ = 6 is  G=                            b0 b1 b2 b3 b4 b5 b6  b6 b0 b1 b2 b3 b4 b5 b5 b6 b0 b1 b2 b3 b4 b4 b5 b6 b0 b1 b2 b3 b3 b4 b5 b6 b0 b1 b2 b2 b3 b4 b5 b6 b0 b1 b1 b2 b3 b4 b5 b6 b0 e e e e e e e             ;             where b0 = 111111, b1 = 100001, b2 = 010010, b3 = 001100, b4 = 001100, b5 = 010010, b6 = 100001, and e = 111111. The code Cp generated by the matrix G is called the NQR Code[110]. If p is of the form 4m + 1 then Tiu and Wallace[110] have shown that dim Cp ≤ p, dH ≥ p − 1, Cp is weakly self dual and that the weight of each codeword is divisible by 4. In the present chapter we show that if p is a prime of the form 4m − 1 then also Cp satisfies similar 59 properties. It is shown that dim Cp = p, Cp is normal and its covering radius is p(p−1) . 2 It is also shown that Cp is Z4 -linear. The following observations about the matrix G are useful for further discussion. Remark 6.1 In R0 , the entries corresponding to the block B0 has all 0’s (all 1’s) if p is of the form 4m + 1 (4m − 1). Remark 6.2 For a > 0, ba = bp−a . Remark 6.3 In any row of G, the entries corresponding to any block Ba , have mirror symmetry with respect to its center. Thus, in Ri , the entry corresponding to point (a, j) is same as the entry corresponding to the point (a, p − j) for every 0 ≤ a ≤ p − 1 and 1≤j≤ (p−1) . 2 The following result of Perrön [85] (also see [79]) about the q.r. is useful in determining properties of NQR codes. Theorem 6.1 (i) Let p = 4m + 1 and let r1 , r2 , . . . , r2m+1 be the (2m + 1) quadratic residues mod p (together with 0). If k is a q.n.r. mod p then among (2m + 1) numbers ri +k there are m quadratic residues (not including 0) and m+1 quadratic non residues. (ii) Let p = 4m + 1 and let n1 , n2 , . . . , n2m be the 2m quadratic non residues mod p. If k is a q.r. mod p then among the 2m numbers ni + k there are exactly m quadratic residues (not including 0) and m quadratic non residues. (iii) Let p = 4m − 1 be a prime and let r1 , r2 , . . . , r2m be the 2m quadratic residues mod p (together with 0). If a is relatively prime to p then among the 2m numbers ri + a there are m quadratic residues and m quadratic non residues. (iv) Let p = 4m − 1. Suppose n1 , n2 , . . . , n2m−1 are the (2m − 1) quadratic non residues and let a be relatively prime to p. Then among the (2m − 1) numbers ni + a there are m quadratic residues (possibly including 0) and m − 1 quadratic non residues. 60 Lemma 6.2 Every row of G except the last row has weight Proof: (p−1)2 . 2 Since each row except the last row of G is a block wise cyclic shift of R0 , wt(Ri ) = wt(R0 ) for all i = 1, 2, . . . , p − 1. By Remark 6.2, wt(R0 ) = wt(b0 ) + 2{wt(b1 ) + . . . + wt(b p−1 )} (6.2) 2 and by Remark 6.1, wt(b0 ) = To determine wt(ba ) for a      0 if p = 4m + 1 p − 1 if p = 4m − 1. 6= 0, let Ba′ = 2 {(a, 1), . . . , (a, (p−1) )}, and let 2 N(Ba′ ) = {a2 − δ.12 , a2 − δ.22 , . . . , a2 − δ. (p−1) }. If p = 4m + 1 by (ii) of 4 Theorem 6.1, N(Ba′ ) contains m quadratic residues and m quadratic non residues. Therefore by equation (6.2), wt(R0 ) = 0 + 2.{2m + . . . + 2m} = 8m2 = (p−1)2 . 2 If p = 4m − 1, by (iii) of Theorem 6.1 there will be (m − 1) quadratic residues and m quadratic non residues in N(Ba′ ). Hence by Equation (6.2) wt(R0 ) = (p−1)2 . 2 As an immediate Corollary we have Corollary 6.3 The minimum distance of Cp is at most Lemma 6.4 Weight of each column of G is Proof: (p−1)2 . 2 (p+1) . 2 It is sufficient to show that the result is true for the coordinate position (0, a) ∈ Hp , for some a ∈ GF (p). The first p entries in this column will be entries from R0 corresponding to coordinate positions Ma = {(0, a), (1, a), . . . , (p − 1, a)}. Let N(Ma ) = {02 −δa2 , . . . , (p−1)2 −δa2 }. If p = 4m+1, then (−δa2 ) is a q.n.r. and hence by (i) of Theorem 6.1, N(Ma ) contains 2m quadratic residues and (2m + 1) quadratic non residues. Hence total number of 1’s in the ath column is (2m + 1). If p = 4m−1, then (−δa2 ) is a q.r. and (p, −δa2 ) = 1. Hence by (iii) of Theorem 6.1, N(Ma ) contains (2m − 1) quadratic residues and 2m quadratic non residues. Thus the 61 number of 1’s in the ath column is 2m. The following relation among the rows of G follows immediately from the above Lemma. Corollary 6.5 (i) (ii) If p = 4m + 1 then R0 = R1 + R2 + . . . + Rp−1 ; If p = 4m − 1 then Rp = R0 + R1 + . . . + Rp−1 . Before stating our next theorem we recall some basic facts needed about the circulant matrices over binary fields as they play an important role in the proof. A binary square matrix C of order p is said to be circulant if C = (cij ) = (cj−i+1) := cir(γ) ; where γ = (c0 , c1 , . . . , cp−1 ) is the first row of C and subscripts are mod p. We state some of its basic properties in the following Lemma. These are given as an exercise in[79]. Lemma 6.6 (i) The algebra of p × p circulant matrices over a field IF is isomorphic to the algebra of polynomials in the ring IF [z]/(z p − 1), if the circulant matrix C = cir(γ) is mapped onto the polynomial Pγ (z) = c0 + c1 z + . . . + cp−1 z p−1 . (ii) Let α0 (= 1), α1 , . . . , αp−1 be the pth roots of unity in some extension field, say, GF (2r ) and let A = (aij ) = (αij ), 0 ≤ i, j ≤ p − 1, be the van der Monde matrix over GF (2r ). Then A diagonalizes C with diagonal entries as the eigen values of C. More precisely we have, A−1 CA = diag(Pγ (α0 ), Pγ (α1 ), . . . , Pγ (αp−1 )). Theorem 6.7 dim Cp = p, (p 6= 2). Proof: Let C be the p × p sub matrix of G obtained by deleting the last row and by considering columns corresponding to coordinate positions (0, 1), (1, 1) (2, 1), . . . , (p − 1, 1). In view of Corollary 6.5 it is sufficient to show that rank(C) is 62 p − 1 or p according to the form 4m + 1 or 4m − 1 of p. We observe that C is a binary circulant matrix, say , C = cir(γ) = cir(c0 , c1 , . . . , cp−1 ); ci ∈ GF (2). Hence, by Lemma 6.6, A−1 CA = diag(Pγ (α0 ), Pγ (α1 ), . . . , Pγ (αp−1 )); where the matrix A, the polynomial Pγ (z) = c0 + c1 z + . . . + cp−1 z p−1 , and the αi ’s for 0 ≤ i ≤ p − 1 are as in Lemma 6.6. If 1 ≤ i ≤ p − 1, Pγ (αi ) 6= 0 and Pγ (α0 ) = p−1 2 ≡ 0(mod2) if and only if p is of the form 4m + 1. Recall that PSL2 (p), the projective special linear group, is the factor group SL2 (p)/ζ, where SL2 (p) is the special linear group on GF (p) and ζ = {I, −I} with I the 2 × 2 identity matrix. Moreover, PSL2 (p) is generated by the coset of matrices   S=  1 1   0 1   On choosing a coset representative  action on a point in Hp by     a b  c d  (x + √   and T =  a b  c d   0 −1  1 0 . for each member we can define the group √ √ a(x + δy) + b √ = x′ + δy ′ . δy) = c(x + δy) + d If p = 4m + 1 then Tiu and Wallace have shown that PSL2 (p) fixes Cp and acts as a group of automorphism for Cp . The following theorem shows that the result is also true for primes of the form 4m − 1. Theorem 6.8 PSL2 (p) fixes Cp and acts transitively on the coordinate positions. Proof: Proof of the fact that P SL2 (p) fixes Cp for p = 4m − 1 is similar to the case √ √ p = 4m + 1 (see[110]). To prove the next part, for given x + δ y and u + δ v in Hp √ √ we have to show that there exist P ∈ PSL2 (p) such that P (x + δ y) = u + δ v. If 63   P1 =   1 −x  0 1    and P2 =   1 u  0 1  √ √ √ √ then P1 (x + δy) = δy and P2 ( δv) = u + δv. Thus we need to find Q ∈ PSL2 (p) such that Q( √ √ δ y) = δ v. Two cases arise: Case 1: Both v and y are either quadratic residues or quadratic non residues mod   p. Then v = a2 y for some a ∈ GF (p). Let Q =   a 0  , √ √ then Q( δy) = v δ. 0 a−1 Case 2: Either v is a q.r. and y is a q.n.r. or v is a q.n.r. and y is a q.r. mod p. In either case v · y · δ is a q.r. mod p and hence v · y · δ = a2 for some a ∈ GF (p). Let   Q= 0  a  a−1 0  √ √ then Q( δy) = v · δ. Theorem 6.9 Cp is Z4 -linear. Proof: It follows from Theorem 2.3 and Remark 6.3. The Covering Radius R(C) of a linear code C is the least integer R such that spheres of radius R around the codewords cover the whole space. It is known that R(C) is the maximum weight of the coset leader. For i = 1, 2, . . . , n and for α ∈ IF 2 let Cα(i) = {(c1 , c2 , . . . , cn ) ∈ C|ci = α} and let N (i) = maxx∈IF n2 { P x∈IF n 2 d(x, Cα(i) )} with the convention that d(x, Φ) = n; where Φ is the empty set. The number N (i) is called the Norm of C with respect to the ith coordinate and N(C) = mini N (i) is called the Norm of the code C. If for some i, N (i) ≤ 2R + 1, then C is said to be normal and the coordinate i is called acceptable. The following theorem determines the covering radius of Cp and shows that Cp is normal. Theorem 6.10 Covering radius of Cp is p(p−1) . 2 Moreover Cp is a normal code with every coordinate acceptable. Proof: Observe that N(Cp ) ≤ p(p − 1). Since Cp has no coordinate identically zero, $ % p(p − 1) N(Cp ) ≤ . R(Cp ) ≤ 2 2 (6.3) 64 p(p−1) Let x ∈ IF2 such that its coordinates when divided in p blocks each of length (p − 1) have a 1 in the first p−1 2 coordinate positions of each block and zero at other positions. Then by Remark 6.2 d(x, c) = (6.3) we get R(Cp ) = p(p−1) . 2 p(p−1) 2 for all c ∈ Cp . Thus using Equation Since N(Cp ) ≤ p(p − 1) < 2R(Cp ) + 1, Cp is normal. Moreover Since Cp is an even code, N(Cp ) = 2R(Cp ) = p(p − 1)[45]. Hence its every coordinate is acceptable. In[66], Janwa has shown that the length n, the minimum distance d , the dimension & ' H k and the covering radius R of a linear code are related by this to the code Cp we get the following corollary. Pk i=1 dH 2i ≤ n−R. Applying Corollary 6.11 The minimum distance d of an NQR code Cp satisfies the inequality p X i=1 & ' p(p − 1) d ≤ i 2 2 The bound given by Corollary 6.11 although poor but is better than the bound given by Corollary 6.3. Chapter 7 Conclusions Anything that won’t sell, I don’t want to invent. Its sale is proof of utility, and utility is success. . . . Thomas A. Edison (1847-1931) In the present dissertation, we have studied Simplex Codes of type α and β, first order Reed Müller code, over Z2s . The concept of 2-dimension has been used extensively to derive the fundamental properties of these codes. It seems to be an appropriate tool for studying linear codes over Z2s . The concept of ‘chain condition’ has been extended in this sense. The method used can be easily extended to study linear codes over Zps . One will need to suitably extend the definition of φG . 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