See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/242923215 On Linear Codes over $${\mathbb{Z}}_{2^{s}}$$ Article in Designs Codes and Cryptography · September 2005 DOI: 10.1007/s10623-004-1717-1 · Source: dx.doi.org CITATIONS READS 4 38 3 authors, including: Manish Kumar Gupta A. K. Lal 43 PUBLICATIONS 154 CITATIONS 45 PUBLICATIONS 181 CITATIONS Dhirubhai Ambani Institute of Information an… SEE PROFILE Indian Institute of Technology Kanpur SEE PROFILE Some of the authors of this publication are also working on these related projects: Optimum Algorithm for solving Graph Isomorphism View project On the search of extremal self-dual codes of length 72 View project All content following this page was uploaded by Manish Kumar Gupta on 31 March 2014. The user has requested enhancement of the downloaded file. On Linear Codes over Z2s Manish K. Gupta∗, Mahesh C. Bhandari† and Arbind K. Lal †. October 23, 2002 Abstract In an earlier paper the authors studied simplex codes of type α and β over Z4 and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over Z2s . The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin / Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over Z2s . The above codes are also shown to satisfy the chain condition. Keywords: Linear codes over rings, Generalized Gray map, Simplex code, Reed-Muller code, p-dimension, Generalized Hamming weights (GHWs), Lee weight, Gray image, Weight distributions. 1 Introduction Codes over Zps have been studied in the early seventies by Blake [4, 5], Spiegel [28] and Priti Shankar [26] etc. But not much attention was paid to such codes in the early eighties [33]. The reason for this seems partly to come from the difficulties arising from the presence of zero divisors in Zps . At the end of eighties Nechaev [21] and later Hammons, Kumar, Calderbank, Sloane and Solé [12] have observed that certain non-linear binary codes with good parameters are ∗ M. K. Gupta is with the Department of Computer Science and Engineering, Arizona State University, Tempe, AZ 85287 USA. E-mail:m.k.gupta@ieee.org. A part of this paper is contained in his Ph.D. Thesis from IIT Kanpur, INDIA. † M.C. Bhandari and A.K. Lal are with the Department of Mathematics, Indian Institute of Technology, Kanpur, INDIA. E-mails: {mcb,arlal}@iitk.ac.in. 1 linear over Z4 . In particular, they have shown that the Kerdock and Preparatalike codes (slightly different from the original Preparata codes but sharing the same distance structure) are images of some linear codes over Z4 that are dual to each other [12]. To show this they have exploited the isometry between ( Zn4 , Lee distance ) and ( Z2n 2 , Hamming distance ). This has motivated a great deal of research in codes over rings (see [6, 14, 15, 27, 32]). Recently in [3], the authors have investigated simplex codes of type α and β over Z4 and determined some fundamental properties. Some known binary linear codes (meeting the Griesmer bound) and nonlinear codes were obtained as Gray image of these codes. For applications, see [2, 10, 18, 20, 23, 29]. A natural question that comes to one’s mind is the extension of these results to codes over Zps . A Major hurdle is the generalization of the “Gray map.” In [24], Ana Sălăgean-Mandache 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 Zsp with the Hamming metric. However there exists an isometry s−1 appeared in [7]. The generalization of these isometries up between Z2s → Z22 to finite chain rings was considered in [11, 14, 22]. More recently an isometry between codes over Z2s and codes over Z4 has been introduced in [31]. In this correspondence, we study basic properties of simplex codes of type α and β and first order Reed-Muller codes over Z2s . The binary linear codes obtained using the binary image meet the Griesmer bound and hence are optimal, and some binary nonlinear uniformly packed codes are obtained. It is observed that type β simplex codes meet the Griesmer bound for codes over rings [27]. Section 2 contains preliminaries and some useful results. Definitions and basic properties of simplex codes of type α and β are given in section 3. Section 4 contains definitions and properties of the first order Reed-Muller codes over Z2s . 2 Preliminaries Calderbank and Sloane [6] have shown that the generator matrix G of any linear code C of length n over Zps is   Ik0 A01 A02 · · · A0s−1 A0s  0 pIk pA12 · · · pA1s−1  pA1s 1    0 p2 Ik2 · · · p2 A2s−1 p2 A2s  G= 0 (1) ,  ..  . .. .. . . .. .. ..  .  . . s−1 s−1 0 0 0 · · · p Iks−1 p As−1s where Aij are matrices over Zps and the columns P are grouped into blockskof sizes k0 , k1 , · · · , ks−1 , ks , respectively. Let k = s−1 i=0 (s − i)ki . Then |C| = p . 2 For x, y ∈ Znq , dH (x, y) = |{i : xi 6= yi }| is called the Hamming distance between x and y and wH (x) = dH (x, 0), the Hamming weight of x. The minimum Hamming distance of C is denoted by dH . The Lee weight [17] of a ∈PZq is given by wL (a) = min {a, q − a} . For x = (x1 , x2 , . . . , xn ) ∈ Znq , wL (x) = ni=1 wL (xi ) and for x, y ∈ Znq , the Lee distance between x and y, denoted, dL (x, y) = wL (x − y). The minimum Lee distance dL of a code C is defined analogously. 2.1 p-dimension of Linear Codes over Zps Let C be a linear code over Zps . Consider the radical series C ⊃ p C ⊃ p2 C ⊃ · · · ⊃ ps−1 C ⊃ ps C = {0} of C. In this series, each associated quotient pl C/pl+1 C, 0 ≤ l ≤ s − 1, is a vector space over the field Zp with basis vl1 + pl+1 C, vl2 + pl+1 C, . . . , vltl + pl+1 C, say. Then the ordered collection B = {v01 , . . . , v0t0 , v11 , . . . , v1t1 , . . . , vs−1 1 , . . . , vs−1ts−1 } ⊂ C satisfies: P λi vi . 1. Given v ∈ C, there exists unique λi ∈ Zp such that v = vi ∈B 2. For any ℓ, 0 ≤ ℓ ≤ s − 1 and any j, 1 ≤ j ≤ tℓ , p vℓj ∈ pℓ+1 C. P λi vi = 0 for 3. Any p−linear combination of vectors in B is zero (i.e., i λi ∈ Zp and vi ∈ B if and only if each λi = 0). Any ordered set D ⊂ C with the above three properties is called a p−basis for the code C and the cardinality of the set D is called the p−dimension of C, denoted p−dim(C). Recently Vazirani, Saran and Sundar Rajan have
used the concept of p−dimension extensively in [30]. Observe that p−dim Znps = sn. For a subset B of C to be a p-basis for C it is necessary and sufficient that every vector in C be a unique p-linear combination of vectors in B. In this correspondence, hv1 , . . . , vk i will denote the p−linear combination of the vectors v1 , . . . , vk . Let   Ik 0 A01 A02 · · · A0s−1 A0s  pIk0  pA01 pA02 · · · pA0s−1 pA0s    0  pIk1 pA12 · · · pA1s−1 pA1s   .. .. .. ..  ..   .  . . . .  .  .. .. .. ..  ..  . . . . . D= (2)  s−1  s−1 s−1 s−1 s−1 p I p A p A · · · p A p A   k0 01 02 0s−1 0s   ps−1 Ik1 ps−1 A12 · · · ps−1 A1s−1 ps−1 A1s   0   0 ps−1 Ik2 · · · ps−1 A2s−1 ps−1 A2s   0  .  .. .. .. . . . ..  ..  . . . . 0 0 · · · ps−1 Iks−1 0 3 ps−1 As−1s Then the rows of D form a p−basis for the code P C having G given in (1) as the generator matrix. Thus, p−dim(C) = k = s−1 i=0 (s − i)ki . The matrix B given below is needed in the proof of Lemma 1 and is row-equivalent to the matrix D (using block permutations applied to its rows):   Ik 0 A01 A02 ··· A0s−1 A0s  pIk0  pA01 pA02 ··· pA0s−1 pA0s   . . . . . .   .. .. .. .. .. ..    s−1  s−1 s−1 s−1 s−1  p Ik0 p A01 p A02 · · · p A0s−1 p A0s      0 pIk1 pA12 ··· pA1s−1 pA1s   B= (3) 0 p 2 Ik 1 p2 A12 · · · p2 A1s−1 p2 A1s  .   .. .. .. .. .. ...   . . . . .    s−1 s−1 s−1 s−1 0 p Ik1 p A12 · · · p A1s−1 p A1s      . . . . . . .. .. .. .. .. ..   s−1 s−1 0 0 0 · · · p Iks−1 p As−1s 2.2 Generalized Gray Map Let Fq be a field with q elements and let u and 1 be the vectors of length q such that u lists all the elements of Fq and 1 is the vector of all 1’s. For a fixed positive integer s, 0 ≤ i ≤ s − 1, define ci := (1 + δi,0 (u − 1)) ⊗ · · · ⊗ (1 + δi,s−2 (u − 1)) where δi,j is the Kronecker delta symbol and ⊗ represents the tensor product (expanded from right to left). Observe that the entries of ci ’s are elements of Fq . Let C = {c0 , c1 , . . . , cs−1 }. It has been observed (see [11]) that C spans a firstorder Reed-Muller code RM (1, s − 1) over Fq . Thus all the codewords of the code spanned by C, except cs−1 and its multiples are permutationally equivalent and have Hamming weight (q − 1)q s−2 . Using C, Greferath and Schmidt [11] have extended the well-known Gray map over Z4 to a finite chain ring R of length s. For a prime p, let Fp be the residue field of R, ν : R −→ Fp the natural surjection, and let T be a set of representative of Fp in R. Then for any r ∈ R, there exist unique ri ∈ T, 0 ≤ i ≤ s − 1 such that r = r0 + r1 p + · · · + rs−1 ps−1 . With the above notations, the generalized Gray map is the bijection γ : R −→ C, r 7−→ ν(r0 )c0 + ν(r1 )c1 + . . . + ν(rs−1 )cs−1 . In particular, for R = Z2s the extended Gray map coincides with that of Carlet’s Generalized Gray map (see [7] ). In this case, the residue field consists of F2 = {0, 1} and hence for 0 ≤ i ≤ s − 1, the entries of ci ’s are either 0 or 1, and γ(1) = c0 , γ(2) = c1 , . . . , γ(2s−1 ) = cs−1 . Thus, for u ∈ Z2s , u 6= 0, one has 4 wt(γ(u)) =  2s−2 , u 6= 2s−1 2s−1 , u = 2s−1 . (4) This weight is (up to a constant factor) the same as the homogeneous weight introduced by Constantinescu, Heise and Honold [9] (see also [16]). Thus this weight will be called homogeneous weight of u and we denote it by wHW (u). From now on we will restrict γ to the case R = Z2s . Also, the Gray isometry from s−1 s−1 Zn2s to Z22 n is the coordinate-wise extension of γ from Z2s to Z22 and with an abuse of notation we call it γ. Definition 1 A binary code is called Z2s -linear if it is permutation equivalent to γ(C) for some linear code C over Z2s . A necessary and sufficient condition for Z8 -linearity is given in [7]. It is also shown that any Z2s -linear code is distance invariant and dH (γ(u), γ(v)) = wHW (u − v) [7]. The minimum homogeneous weight, dHW , of C can be defined in the usual sense. Note that for s = 2, dHW = dL . To state the next lemma (to be used in Section 3.1), we need the following notation. For a fixed integer s define Gs−j := (cj , cj+1 , . . . , cs−1 )t for 0 ≤ j ≤ s−1. Note that in this notation, Gs is a generator matrix of the binary first-order Reed-Muller code RM (1, s − 1). Lemma 1 For p = 2, the images of the rows of the matrix B given by (3) under the generalized Gray map γ are linearly independent over Z2 . Proof. Applying the generalized Gray map γ to the rows of B yields G s ⊗ Ik0 ∗ ··· ∗ ∗  Gs−1 ⊗ Ik1 · · · ∗ ∗    γ(B) =  .. .. .. ..  . ...  . . . .  · · · G1 ⊗ Iks−1 ∗ Since the Gi ’s for 1 ≤ i ≤ s are linearly independent, the result follows. Remark 1 Lemma 1 implies ‘the image set of a 2-basis of C under the generalized Gray map gives rise to a linearly independent set over Z2 ’. The converse of this need not be true in general. For example, If s = 2, then {(1011), (0111)} is Z2 -independent but {(3 2),(1 2)} is not. A linear code C over Z2s of length n, 2-dimension k, minimum Hamming distance dH , minimum Lee distance dL , and minimum homogeneous distance dHW is called an [n, k, dH , dL , dHW ] code or simply an [n, k] code. Note that, for an [n, k] linear code C over Zps , the dual code C ⊥ is an [n, sn−k] linear code. C is called Z2 -linear if γ(C) is a binary linear code. Thus, if C is an [n, k, dH , dL , dHW ] Z2 -linear code then γ(C) has parameters [2s−1 n, k, dHW (C)]. 5 2.3 GHWs of Linear Codes over Zps Let C be an [n, k] linear code over Zps . The definitions of Generalized Hamming weights (GHWs), weight hierarchies and chain condition given in [13] 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 the word ‘dimension’ by ‘p-dimension’ everywhere. Thus for 1 ≤ r ≤ k, the rth 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 the basic properties of GHW’s. Theorem 1 [1] Let C be an [n, k] linear code over Zps and C ⊥ be the dual code defined with respect to the standard inner product in 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): The weight hierarchies of C and its dual code C ⊥ are related by {dr (C) | 1 ≤ r ≤ k} = {1, · · · , 1, . . . , n, · · · , n}\{n + 1 − dr (C ⊥ ) | 1 ≤ r ≤ sn − k}. | {z } | {z } s s The next lemma connects the support size of a code with its Lee and homogeneous weights. Lemma [n, r] linear code overPZ2s then P 2 If D is an r+s−2 r+s−2 1. wS (D) and 2. wS (D). c∈D wL (c) = 2 c∈D wHW (c) = 2 Proof. Consider the (2r × n) matrix consisting of all codewords in D. For 0 ≤ m ≤ s, let nm−1 be the number of columns that containPthe entries 0, 1 · 2s−m , 2 · 2s−m , 3 · 2s−m , · · · , (2m − 1) · 2s−m equally often. Then sm=1 nm−1 = wS (D) and since there are 2r rows, one has X wL (c) = c∈D = s  X m=1 s X m=1 2r nm−1 m 2 (2m −1) X t=0 s−m wL (t · 2 
) (5)
nm−1 2r−m (2s+m−2 ) = 2r+s−2 · wS (D)  P m −1) Pm as (2 wL (t · 2s−m ) = 2s−m 2t=1−1 min t, 2m − t = 2s+m−2 . This completes t=0 part 1. For part 2, replace by homogeneous (5) P2m −1 Lee weight P weight in equation s−m s−1 s−m and use (4) to get w (t · 2 ) = 2 + w (t · 2 ) = HW t6=2m−1 HW
s−2 t=0 s−1 m s+m−2 2 + 2 −2 2 =2 . 6 Remark 2 For s = 2, Lemma 2 was first proved by K. Yang et al in [34] and later it has been done in higher generality by Constantinescu et al in [9]. Corollary 1 (Plotkin-Type Bound) Let C be an [n, k, dL , dHW ] code over Z2s . k+s−2 k+s−2 Then dL ≤ 2(2k −1)n and dHW ≤ 2(2k −1)n . Proof. Every nonzero codeword has Lee weight at least dL . Thus the total sum of Lee weights of codewords is at least (2k − 1)dL and an upper bound for the total sum of Lee weights of codewords is 2k+s−2 n. Therefore (2k −1)dL ≤ 2k+s−2 n. A similar argument holds for the case of homogeneous weights. The proof of the following corollary follows immediately from Lemma 2. Corollary 2nIf 1 ≤ r ≤ k, then the o rth GHW of C satisfy r r −1)dL HW ⌉, ⌈ (2 2−1)d ⌉ . dr (C) ≥ max ⌈ (22r+s−2 r+s−2 Remark 3 Using Lemma 2, for 1 ≤ r ≤ k, the rth GHW can also be defined as  P 1 dr (C) = 2r+s−2 min d∈Dr wL (d) | Dr is an [n, r] subcode of C . Similar definition holds for wHW . If r = 1, we get Corollary 3 from Lemma 2.  L ⌉, ⌈ d2HW . Corollary 3 Let C be a linear code over Z2s , then dH ≥ max ⌈ 2ds−1 s−1 ⌉ Definition 2 A linear code C over Z2s is said to be of type α (β) if     dHW dHW dH = dH > . 2s−1 2s−1 3 Z2s -Simplex Codes of Type α and β Let Gαk be a k × 2sk matrix over Z2s with Gα1 = [0 1 2 3 · · · (2s − 1)], and for (0 1 2 3 · · · (2s − 1)) ⊗ 1 , where 1 (the all 1 vector) in the k ≥ 2, Gαk = 1 ⊗ Gαk−1 first row is of length 2s(k−1) and that in the second row is of length 2s . Clearly, the code Skα generated by Gαk over Z2s has length 2sk and 2-dimension sk. The following remarks are straight forward. Remark 4 If Ak−1 denotes the (2s(k−1) × 2s(k−1) ) array consisting of all codeα words in Sk−1 and if J is the matrix of all 1′ s then the (2sk × 2sk ) array of codewords of Skα is given by 7        Ak−1 Ak−1 Ak−1 .. . Ak−1 J + Ak−1 2J + Ak−1 .. . Ak−1 2J + Ak−1 22 J + Ak−1 .. . ··· Ak−1 s · · · (2 − 1)J + Ak−1 · · · (2s+1 − 2)J + Ak−1 .. ... . Ak−1 (2s − 1)J + Ak−1 (2s+1 − 2)J + Ak−1 · · · (2s − 1)2 J + Ak−1     .   Remark 5 If R1 , R2 , ..., Rk denote the rows of the matrix Gαk then wH (Ri ) = 2sk − 2s(k−1) , wH (2s−1 Ri ) = 2sk−1 , wL (Ri ) = 2s(k+1)−2 , and wHW (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 Gαk . Writing 2s in place of 2s+j for j > 0 (as 2s ≡ 2s+j (mod 2s )), we have the following lemma. Lemma 3 Let c ∈ Skα be a type m codeword. Then all the components of c will occur equally often 2sk−m times. α s α Proof. Any x ∈ Sk−1 gives
rise to the following 2 codewordss of Sk
y1 = x x x · · · x , y2 = x 1 + x
2 + x · · · (2 − 1) + x , x 2 + x 22 + x · · · (2s − 2) + x , . . .
, and y3 = y2s = x (2s − 1) + x (2s − 2) + x · · · 1 + x , where i = (i i · · · i). Hence the proof follows easily by induction on k and the Remark 4. 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 t Cm = [2s−m R1t , . . . , 2s−1 R1t , . . . , 2s−m Rkt , . . . , 2s−1 Rkt ] , where Ri is the ith row of the matrix Gαk and the superscript t denotes the transpose. The subcodes D(m) of C generated by the 2-linear combinations of the rows of Cm will have 2mk codewords. Note that out of these codewords we get a codeword of type m duplicated 2(m−1)k times. 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 Let 0 < m ≤ s. Then the number of codewords of type m in Skα is 2(m−1)k (2k − 1). Theorem 2 The Hamming, Lee and homogeneous weight distributions of Skα are: 1. AH (0) = 1, AH (2sk−m (2m − 1)) = 2(m−1)k (2k − 1) for 1 ≤ m ≤ s, 8 2. AL (0) = 1, AL (2s(k+1)−2 ) = 2sk − 1 and 3. AHW (0) = 1, AHW (2s(k+1)−2 ) = 2sk − 1. Proof. Let c ∈ Skα be a codeword of type m (6= 0). Then by Lemma 3, wH (c) = (m−1)k k 2sk − 2sk−m and thus by Lemma 4, AH (2sk−m (2m − 1)) (2 − 1). For P= m2 (2 −1) sk−m m = 0, AH (0) = 1. Also, by Lemma 3 wL (c) = 2 wL (t · 2s−m ) = t=0 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 homogeneous weight. Remark 6 1. Skα is an equidistant code with respect to Lee and homogeneous distances. The study of unicity as a equidistant code can be done as it was done for Z4 linear codes by Carlet in [8]. 2. Skα is of type α. 3. For s = 1, Skα reduces to an extended binary simplex code Sˆk . 4. The Lee weight distribution of the code Skα was first determined by Satyanarayna [25]. 5. The minimum weights of Skα are dH = 2sk−1 and dL = dHW = 2s(k+1)−2 .   1 (0 2 4 6 · · · (2s − 2)) β β Let Gk be the k × n(k) matrix with G2 = , Gα1 1   (0 2 4 6 · · · (2s − 2)) ⊗ 1 1 , where Gαk−1 is the and for k > 2, Gβk = α Gk−1 1 ⊗ Gβk−1 α generator matrix of Sk−1 and n(k) is the length of the linear code Skβ generated by Gβk over Z2s . Here all the five 1’s are of appropriate sizes. The definition of the matrix Gβ2 gives n(2) = 3 · 2s−1 and the structure of the matrix Gβk with an inductive argument implies n(k) = 2(s−1)(k−1) (2k − 1). The two rows of Gβ2 generate a free module and the same is true of the k rows of Gβk (using an induction argument). Thus the 2-dimension of Skβ is sk. We now show by induction on k, that no two columns of Gβk are multiples of each other. It is clearly true for k = 2. Let the result be true for any two columns of Gβk−1 . Let c1 and c2 be any two columns of Gβk . Observe that Gβk can be split into (2s−1 + 1) blocks as B0 , B1 , B2 , . . . , B2s−1 with the corresponding first row of Gβk as (1 · · · 1 | 0 · · · 0 | 2 · · · 2 | · · · | (2s − 2) · · · (2s − 2)). When c1 , c2 ∈ Bi (1 ≤ i ≤ 2s−1 ) they are not multiples by induction hypothesis. If c1 , c2 ∈ B0 then they are not multiples as each column of Gαk−1 is distinct and the top row of B0 is the all 1 vector. Now let c1 ∈ B0 and c2 ∈ Bi (1 ≤ i ≤ 2s−1 ). 9 Then for λ ∈ Z2s , λc2 = z  for some zero divisor z. Thus λc2 6= c1 . Finally ⋆ let c1 ∈ Bi and c2 ∈ Bj (1 ≤ i 6= j ≤ 2s−1 ). Then c1 and c2 are not multiples as their first entries can be multiples of each other, but then there  (s−1)(k−1)  is at least one 1 β k in the remaining entries. Thus Sk is a 2 (2 − 1), sk code. Remark 7 If Ak−1 denotes the (2s(k−1) × 2s(k−1) ) array consisting of all codeα words in Sk−1 , Bk−1 denotes the 2s(k−1) × 2(s−1)(k−2) (2k−1 − 1) array of codewords β in Sk−1 and if J is the matrix of all 1′ s then the 2sk × 2(s−1)(k−1) (2k − 1) array of all codewords of Skβ is given by        Ak−1 J + Ak−1 2J + Ak−1 . . . s (2 − 1)J + Ak−1 Bk−1 Bk−1 Bk−1 . . . Bk−1 Bk−1 2J + Bk−1 22 J + Bk−1 . . . s (2 − 2)J + Bk−1 Bk−1 + Bk−1 23 J + Bk−1 . . . s 2 (2 − 2 )J + Bk−1 22 J ··· ··· ··· .. . ··· Bk−1 − 2)J + Bk−1 (2s − 22 )J + Bk−1 . . . 2J + Bk−1 (2s     .   Let U , Z be the set of units and zero divisors of Z2s , respectively. Then the following propositions help in determining weight distributions of Skβ . Proposition 1 For 1 ≤ j ≤ k, let Rj be the j th row of Gβk . Then P s(k−1) 1. and each zero divisor in Z2s occurs 2(s−1)(k−2) (2k−1 − 1) i∈U ωi = 2 times in Rj .   2. wH (Rj ) = 2(s−1)(k−1)−s (2k − 1)(2s − 1) + 1 , wHW (Rj ) = 2sk−k−1 (2k − 1) for all j, 1 ≤ j ≤ k. 3. wL (R1 ) = 2s(k−1) + 2sk−2 − 2sk−k−1 . Proof. The proof follows directly from above using the definition of Rj . The next proposition gives the structure of codewords of Skβ . P Proposition 2 Let c ∈ Skβ . If one of the coordinates of c is a unit then i∈U ωi = 2s(k−1) and each zero divisor in Z2s occurs 2(s−1)(k−2) (2k−1 − 1) times in c. β α Proof. By Remark 7, there exist y1 ∈ Sk−1 and y2 ∈ Sk−1 such that c can have any of the following 2s forms

c = y1 y2 y2 · · · y2 , c = 1 + y1 y2
2 + y2 · · · (2s − 2) + y2 , c = 2 + y1 y2 22 + y2 · · · (2s − 22 ) + y2 , . . .
, or c = (2s − 1) + y1 y2 (2s − 2) + y2 · · · 2 + y2 , where i = (i i . . . i). The proof now follows by induction. 10 s−1 Let C be a linear code over for all i} .  Z2s and D = {c ∈ C | ci = 0 or 2 1 Then C (2) = c | c ∈ D is called the torsion code of C and the binary code 2s−1 C (1) = C (mod 2) is called the reduction code of C. If C is  a free module then (2) (1) α C = C . Hence the reduction and torsion codes of Sk Skβ are equal. The next proposition determines these binary codes. Proposition 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 follows by induction on k and is similar to the proof given for codes over Z4 (see [3] ). Theorem 3 The Hamming and homogeneous 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), 1 ≤ m ≤ s and 2. AHW (0) = 1, AHW (2sk−1 ) = 2k − 1, AHW (2sk−k−1 (2k − 1)) = 2k (2(s−1)k − 1). Proof. Using Theorem 2, Remark 7 and induction on k, the possible nonzero Hamming (homogeneous) weights of Skβ are {2
(s−1)(k−1) (2k−m (2m − 1)+ (21−m − 1)) | 1 ≤ m ≤ s} 2sk−1 , 2sk−k−1 (2k − 1) . By Lemma 4, a Hamming weight of type m occurs 2(m−1)k (2k − 1) times. Moreover homogeneous weight 2sk−1 occurs 2k − 1 times. Thus the other weight occurs 2sk − 2k times. The next corollary follows directly from Theorem 3. Corollary 4 1. Skβ is of type β. 2. For s = 1, Skβ reduces to the binary simplex code Sk . 3. dH (Skβ ) = 2s(k−1) and dHW (Skβ ) = 2sk−k−1 (2k − 1). Recently a Griesmer bound for codes over finite quasi-Frobenius rings has been obtained by Shiromoto and Strome in [27]. In particular, they prove: Theorem 4 [27] For a free linear code C of length n, dimension k and minimum Hamming distance dH over Zps the following inequality holds:  k−1  X dH . n≥ pi i=0 Applying the above inequality to Skβ for p = 2 yields: Proposition 4 The simplex codes of type β meet the Griesmer bound for codes over Zps . 11 3.1 Gray Image Families In this section we study the Gray images of the above simplex codes in two ways. The first method consists of applying the generalized Gray map defined in section 2.2 to the linear [n, k, dH , dHW ] code C over Z2s . This code, denoted γ(C), is (possibly) non-linear, is of length 2s−1 n, has minimum Hamming distance dHW (since for any Z2s linear code, dH (γ(u), γ(v)) = wHW (u − v) = dHW (u, v) [7]), and also has 2k codewords. The second method consists of applying the generalized Gray map γ to the matrix of the 2−basis of C. This matrix has full row-rank (see Lemma 1) and hence can be taken as the generator matrix of a binary linear code. The code obtained by the second method also has length 2s−1 n and dimension k. It will be denoted by Cγ . For example, let B be the matrix given  in (3) for  p = 2. The rows of B form dHW a 2-basis for C. Then the code Cγ is a 2s−1 n, k, ≥ binary linear code. 2s−1 Note that γ(C) and Cγ have the same number of codewords but they are not equal in general. In this section, we study γ(C) and Cγ codes for the simplex codes of type α and β and also determine their weight hierarchies. Let S¯kα be the punctured code of Skα obtained by deleting the zero coordinate. Then the following remark is immediate. Remark 8 (i) γ(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. [19]) and nh2dH . (ii) γ(Skβ ) is a binary code of length 2k(s−1) (2k − 1) and minimum Hamming distance 2sk−k−1 (2k − 1). It is an example of a code having n = 2dH (cf. [19]). The next two results determine the binary linear codes generated by the generalized Gray map of the 2-basis S¯kα and Skβ .   Theorem 5 Let C = S¯kα . Then Cγ 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 distribution the same as the homogeneous weight distribution of S¯kα . Proof. By Lemma 1, Cγ is a binary linear code of length 2s−1 (2sk − 1) and dimension sk. Let Gkα be a generator matrix of Skα in 2-basis form. Then (0 1 2 3 . . . (2s − 1)) ⊗ 1 2(0 1 2 3 . . . (2s − 1)) ⊗ 1   .  .  .   2s−1 (0 1 2 3 . . . (2s − 1)) ⊗ 1  = 1 ⊗ Gα  k−1  1 ⊗ 2Gα  k−1   . .  .  Gkα 1 ⊗ 2s−1 Gα k−1    γ((0 1 2 3 . . . (2s − 1))) ⊗ 1   γ(2(0 1 2 3 . . . (2s − 1))) ⊗ 1       γ(22 (0 1 2 3 . . . (2s − 1))) ⊗ 1  α) =  & γ(G .   k ..       γ(2s−1 (0 1 2 3 . . . (2s − 1))) ⊗ 1   α ) 1 ⊗ γ(Gk−1       .    The proof is by induction on k. For k = 2, the result follows trivially. Assume 12 α the result holds for k − 1, i.e., γ(Gk−1 ) yields a binary code in which every sk−2 non-zero codeword is of weight 2 . Then by the induction hypothesis the possible non-zero weight from the lower portion of the matrix γ(Gkα ) will be 2s · 2sk−2 = 2s(k+1)−2 . From the structure of the first s rows of γ(Gkα ) 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 Gkα and rearranging the rest of the columns yields the code having 2s−1 copies of Ssk . Theorem 6 Let C = Skβ . Then Cγ is the binary MacDonald code Msk,(s−1)k , with parameters [2sk − 2(s−1)k , sk, 2sk−1 − 2(s−1)k−1 ] and Hamming weight distribution same as the homogeneous weight distribution of Skβ . Proof. By induction on k. Let Gkβ be a generator matrix of Skβ in 2-basis form then   γ(1) γ((0 2 4 6 . . . (2s − 2))) ⊗ 1  γ(21) γ(2(0 2 4 6 . . . (2s − 2))) ⊗ 1      .. .. β γ(Gk ) =  , . .    γ(2s−1 1) γ(2s−1 (0 2 4 6 . . . (2s − 2))) ⊗ 1  β α γ(Gk−1 ) 1 ⊗ γ(Gk−1 ) α α where Gk−1 is the generator matrix of Sk−1 in 2-basis form. It is easy to verify that β β the result holds for k = 2. Assume that result holds for Sk−1 . Then γ(Gk−1 ) yields a binary code with possible non-zero weights either 2sk−s−1 or 2sk−s−k (2k−1 − 1). α By Theorem 5, γ(Gk−1 ) is a binary code in which every non-zero codeword is sk−2 of weight 2 . Thus possible non-zero 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 in the lower portion of the matrix does not change. Note that the codes Cγ of S¯kα and Skβ are binary linear codes meeting the Griesmer bound hence are optimal. For another representation of MacDonald codes see [15]. Finally, the weight hierarchy of Skα and Skβ are given by the following two theorems. TheoremP7 Skα satisfies the chain condition and its weight hierarchy is given by dr (Skα ) = ri=1 2sk−i = 2sk − 2sk−r , 1 ≤ r ≤ sk. Proof. By Remark 6, any r-dimensional subcode of Skα is of constant homogeneous weight. Hence by definition (see Remark 3) 13 dr (Skα ) = 1 2r+s−2 X wHW (c) = c(6=0)∈Dr 2sk+s−2 2r+s−2 X 1 = 2sk−r (2r − 1). Let c(6=0)∈Dr D1 = h2s−1 R1 i, D2 = h2s−1 R1 , 2s−1 R2 i, . . . , Dk = h2s−1 R1 , . . . , 2s−1 Rk i, Dk+1 = h2s−2 R1 , 2s−1 R1 , 2s−1 R2 , . . . , 2s−1 Rk i, . . . , and Dsk = hR1 , 2R1 , . . . , 2s−1 R1 , . . . , Rk , 2Rk , . . . , 2s−1 Rk i. Then D1 ⊆ D2 ⊆ · · · ⊆ Dsk and for 1 ≤ r ≤ sk, wS (Dr ) = dr (Skα ). Theorem 8 For 1 ≤ i ≤ k, (i − 1)s < r ≤ is and n(k) = 2(s−1)(k−1) (2k − 1), dr (Skβ ) = n(k)−2(s−1)(k−1) (2k−r −2i−r ). Moreover Skβ satisfies the chain condition. Proof. The proof follows by induction on k. Clearly the result holds for k = β 2. Assume that the result holds for Sk−1 . Hence, for 1 ≤ i ≤ k − 1, there β exists an r-dimensional subcode of Sk−1 with minimum support size n(k − 1) − 2(s−1)(k−2) (2k−1−r − 2i−r ). By Remark 7, β α dr (Skβ ) = 2s−1 dr (Sk−1 ) + dr (Sk−1 ). (6) α But all r-dimensional subcodes of Sk−1 have constant support size (2sk−s − 2sk−s−r ). Thus simplifying (6) yields the result. The case i = k is trivial. Let D1 Ds+1 D2s Dsk = = = = h2s−1 R1 i, D2 = h2s−2 R1 , 2s−1 R1 i, . . . , Ds = hR1 , 2R1 , 22 R2 , . . . , 2s−1 R1 i, hR1 , 2R1 , 22 R2 , . . . , 2s−1 R1 , 2s−1 R2 i, . . . , hR1 , 2R1 , 22 R2 , . . . , 2s−1 R1 , R2 , 2R2 , . . . , 2s−1 R2 i, . . . , and hR1 , 2R1 , 22 R2 , . . . , 2s−1 R1 , R2 , 2R2 , . . . , 2s−1 R2 , . . . , Rk , 2Rk , 22 Rk , . . . , 2s−1 Rk i. Now D1 ⊆ D2 ⊆ · · · ⊆ Dsk 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 1, 7 and 8. The code γ((Skβ )⊥ ) is a uniformly packed code as it meets the Johnson bound [19]. There exist other type α and type β codes over Z2s . For example, the first order Reed-Muller code R1,m−s+1 defined in the next section is of type α and the upliftred extended Hamming code H2s of length 8 in [6] is of type β. 4 First Order Reed-Muller Code over Z2s In [12], Hammons et al have constructed a linear code over Z4 (called a quaternary first order Reed-Muller code) whose Gray image is the binary first order 14 Reed-Muller code. In this section we construct a linear code over Z2s whose image under the generalized Gray map γ is the binary first order Reed-Muller 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 . Let G be a (m − s + 2) × 2m−s+1 matrix given by (consisting of the rows Z22 as 1 and 2s−1 vi (1 ≤ i ≤ m − s + 1))   0 0 · · · 0 0 2s−1 2s−1 · · · 2s−1 2s−1  .. .. .. .. .. .. ..  . . . .. ...  . . . . . . .  G =  . s−1 (7)  s−1 s−1 s−1  0 2  ··· 0 2 0 2 ··· 0 2 1 1 ··· 1 1 1 1 ··· 1 1 The code generated by G is called the first order Reed-Muller code over Z2s , denoted R1,m−s+1 . It is a [2m−s+1 , m + 1, 2m−s , 2m−s+1 , 2m−1 ] type α linear code over Z2s . Its generator matrix in 2-basis is given by   0 0 ··· 0 0 2s−1 2s−1 · · · 2s−1 2s−1 ..  .. .. .. .. .. .. ... ...  .. .  . . . . . .  .  s−1 s−1 s−1 s−1  ··· 0 2 0 2 ··· 0 2   0 2   1 ··· 1 1 1 1 ··· 1 1  (8) G= 1   2 ··· 2 2 2 2 ··· 2 2   2  . . . . . . . ..  . .  .. .. .. .. .. .. .. .. .. .  s−1 s−1 s−1 s−1 s−1 s−1 s−1 s−1 2 2 ··· 2 2 2 2 ··· 2 2 Remark 9 For s = 2, R1,m−s+1 reduces to the quaternary first order ReedMuller code ZRM (1, m) defined in Hammons et al [12]. In [10], Davis and Jedwab have also introduced two more generalizations of Reed-Muller codes, viz ZRM2s (r, m) and RM2s (r, m). We now find weight distributions of the code R1,m−s+1 . Proposition 5 The Hamming and homogeneous weight distributions of R1,m−s+1 are: 1. AH (0) = 1, AH (2m−s ) = 2m−s+2 − 2 and AH (2m−s+1 ) = 2m+1 − 2m−s+2 + 1, 2. AHW (0) = 1, AHW (2m ) = 1 and AHW (2m−1 ) = 2m+1 − 2. Proof. Note that first s rows of the matrix G given in (8) have Hamming weight 2m−s+1 and remaining m − s + 1 rows are of Hamming weight 2m−s . It is easy to see that any non-trivial 2-linear combination of the last m − s + 1 rows also has Hamming weight 2m−s . Hence AH (2m−s ) = 2m−s+2 − 2. Since −2s−1 = 2s−1 15 in Z2s , every other non-trivial 2-linear combination has weight 2m−s+1 . Thus AH (2m−s+1 ) = 2m+1 − 2m−s+2 + 1. Similar arguments hold for homogeneous weight. Theorem 9 The weight hierarchy of R1,m−s+1 is given by  Pt−1 m−s−i , 1≤t≤m−s+1 1,m−s+1 i=0 2 dt (R )= 2m−s+1 , m − s + 1 < t ≤ m + 1. Moreover, R1,m−s+1 satisfies the chain condition. Proof. By Corollary 2, 1,m−s+1 dt (R  (2t − 1)2m−1 )≥ 2t+s−2  =  Pt−1 m−s−i , i=0 2 m−s+1 2 , 1≤t≤m−s+1 m − s + 1 < t ≤ m + 1. (9) Let 1 ≤ t ≤ m − s + 1 and let D be a t-dimensional subcode of R1,m−s+1 generated by any t rows chosen from the last m − s + 1 rows of (8). Then the chosen tProws share 2m−s+1−t common zero bit positions. Hence the support size m−s−i of D is t−1 . If t > m − s + 1 then trivially equality holds in (9). Suppose i=0 2 D1 Dm−s+1 Dm−s+2 Dm−1 = = = = h2s−1 vm−s+1 i, D2 = h2s−1 vm−s , 2s−1 vm−s+1 i, . . . , h2s−1 v1 , . . . , 2s−1 vm−s , 2s−1 vm−s+1 i, h2s−1 , 2s−1 v1 , . . . , 2s−1 vm−s , 2s−1 vm−s+1 i, . . . , and h1, 2, 22 , . . . , 2s−1 , 2s−1 v1 , . . . , 2s−1 vm−s , 2s−1 vm−s+1 i. Then D1 ⊆ D2 . . . ⊆ Dm+1 and ws (Dr ) = dr (R1,m−s+1 ), 1 ≤ r ≤ m + 1. The map γ is non-linear over Z2s as γ(2 + 3) 6= γ(2) + γ(3). However we have the following lemma. Lemma 5 Let T = {2i : 0 ≤ i ≤ s − 1} ∪ {0}. Then γ(a + b) = γ(a) + γ(b), for all a, b ∈ T. Proof. The proof follows by the definition of the map γ. Theorem 10 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 in (8). Since γ maps G to a generator matrix of a binary first order Reed-Muller code (with some permutation of the rows (see (8) and Lemma 5). Thus γ(c) belongs to the binary linear first order Reed-Muller code. Hence R1,m−s+1 is Z2 -linear. The results presented in this correspondence can be generalized to codes over Zps and will be reported elsewhere. 16 Acknowledgements The authors thank D. G. Glynn, T. A. Gulliver and Patrick Solé for their helpful remarks. References [1] A. E. Ashikhmin, “On generalized hamming weights for galois ring linear codes,” Designs, Codes and Cryptography, vol. 14, 1998, pp. 107–126. [2] E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York, 1968. [3] M. C. Bhandari, M. K. Gupta, and A. K. Lal, “On Z4 simplex codes and their gray images,” AAECC-13, Lecture Notes in Computer Science, vol. 1719, 1999, pp. 170–180. [4] I. F. Blake, “Codes over certain rings,” Inform. Control, vol. 20, 1972, pp. 396–404. [5] I. F. Blake, “Codes over integer residue rings,” Inform. Control, vol. 29, 1975, pp. 295–300. [6] A. R. Calderbank and N. J. A. Sloane, “Modular and p-adic cyclic codes,” Designs, Codes and Cryptography, vol. 6, 1995, pp. 21–35. [7] C. Carlet, “ Z2k -linear codes,” IEEE Trans. Inform. Theory, vol. 44, no. 4, 1998, pp. 1543–1547. [8] C. Carlet, “ One weight Z4 -linear codes ” Int. Conf. on Coding, Crypto and related Areas, Mexico Springer Lecture Notes in Computer Science, 1999, pp. 57–72. [9] I. Constantinescu, W. Heise, and T. Honold “ Monomial extensions of isometries between codes over Zm ” Proc. Workshop ACCT’96, Sozopol, Bulgaria, 1996, pp. 98–104. [10] J. A. Davis and J. Jedwab, “Peak to mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes,” IEEE Trans. Inform. Theory, vol. 45, no. 7, 1999, pp. 2397–2417. [11] M. Greferath, and S. E. Schmidt, “Gray isometries for finite chain rings and a nonlinear ternary (36, 312 , 15) code,” IEEE Trans. Inform. Theory, vol. 45, no. 7, 1999, pp. 2522–2524. 17 [12] A. Roger Hammons, P. Vijay Kumar, A. R. Calderbank, N. J. A. Sloane, and Patrick Solé, “The Z4 -linearity of Kerdock, Preparata, Goethals, and related codes,” IEEE Trans. Inform. Theory, vol. 40, no. 2, 1994, pp. 301– 319. [13] T. Helleseth, T. Klove, V. I. Levenshtein, and φ. Ytrehus, “Bounds on the minimum support weights,” IEEE Trans. Inform. Theory, vol. 41, no. 2, 1995, pp. 432–439. [14] T. Honold, and I. Landjev, “Linearly representable codes over chain rings,” Abh. Math. Sem. Univ. Hamburg, vol. 69, 1999, pp. 187–203. [15] T. Honold, and I. Landjev, “Linear codes over finite chain rings,” Electronic Journal of Combinatorics, vol. 7, no. 1, 2000, Research Paper 11. [16] I. Konstantinesku, and V. Khauıze, “A metric for codes over residue class rings of integers,” Problemy Peredachi Informatsii, vol. 33, no. 3, 1997, pp. 22–28. [17] C. Y. Lee, “Some properties of non-binary error-correcting codes,” IEEE Trans. Inform. Theory, vol. 4, 1958, pp. 77–82. [18] F. J. MacWilliams, “Error correcting codes for multi-level transmission,” Bell System Technical Journal, 1961, pp. 281–308. [19] F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, North Holland, Amsterdam, 1977. [20] J. L. Massey and T. M. Mittelholzer, “Convolutional codes over rings,” in Proc. Joint Swedish-USSR Int. Workshop in Inform. Theory, Gotland, Sweden, 1989, pp. 14–18. [21] A. A. Nechaev, “Kerdock code in a cyclic form,” Discrete Math. Appl., vol. 1, 1991, pp. 365–384. [22] A. A. Nechaev and A. S. Kuzmin, “Linearly presentable codes,” Proc. IEEE Int. Sympos. Inf. Theory Appl., 1996, pp. 31–34. [23] E. E. Nemirovskiy, “Codes on residue class rings with multi-frequency phase telegraphy,” Radiotechnika I electronika, vol. 9, 1984, pp. 1745–1753. [24] Ana Sălăgean-Mandache, “On the isometries between Zpk and Zkp ,” IEEE Trans. Inform. Theory, vol. 45, no. 6, 1999, pp. 2146–2148. [25] C. Satyanarayana, “Lee metric codes over integer residue rings,” IEEE Trans. Inform. Theory, vol. 25, no. 2, 1979, pp. 250–254. 18 [26] Priti Shankar, “On BCH codes over arbitrary integer rings,” IEEE Trans. Inform. Theory, vol. 25, no. 4, 1979, pp. 480–483. [27] K. Shiromoto and L. Storme “ A griesmer bound for codes over finite quasi-frobenius rings” Proc. Workshop WCC 2001, Int. workshop in coding and Cryptography, January 8–12, 2001, Paris, France, to appear in Discr. Applied Math. [28] Eugene Spiegel, “Codes over Zm ,” Inform. Control, vol. 35, 1977, pp. 48–51. [29] B. SundarRajan, Transform Domain Study of Cyclic and Abelian Codes over Residue Class Integer Rings, Ph.D. thesis, Deptt. of Elec. Engg., IIT Kanpur, Kanpur, India, 1989. [30] V. V. Vazirani, H. Saran, and B. SundarRajan, “An efficient algorithm for constructing minimal trellises for codes over finite abelian groups,” IEEE Trans. Inform. Theory, vol. 42, no. 6, 1996, pp. 1839–1854. [31] G. Vega and H. Tapia-Recillas, “On Z2s -linear and quaternary codes,” IEEE Trans. Inform. Theory, (submitted), 2001. [32] Z.X. Wan, Quaternary Codes, World Scientific, Singapore, 1997. [33] Siri K. Wasan, “On codes over Zm ,” IEEE Trans. Inform. Theory, vol. 28, no. 1, 1982, pp. 117–120. [34] K. Yang, T. Helleseth, P. V. Kumar, and A. G. Shangbhag, “On the weight hierarchy of Kerdock codes over Z4 ,” IEEE Trans. Inform. Theory, vol. 42, no. 5, 1996, pp. 1587–1593. 19 Keywords: Linear codes over rings, Generalized Gray map, Simplex code, Reed-Muller code, p-dimension, Generalized Hamming weights (GHWs), Lee weight, Gray image, Weight distributions. 20 Contact Author Manish K. Gupta Room GWC 354, Department of Computer Science and Engineering, College of Engineering and Applied Sciences, Arizona State University, Tempe Arizona, USA 85287-5406 Telephone: +480-965-2776 Fax: +480-965-2751 E-mail: manish.gupta@asu.edu,m.k.gupta@ieee.org 21