On Senary Simplex Codes Manish K. Gupta1 , David G. Glynn1 , and T. Aaron Gulliver2 1 2 Department of Mathematics, University of Canterbury, Private Bag 4800, Christchurch, New Zealand m.k.gupta@ieee.org d.glynn@math.canterbury.ac.nz Department of Electrical and Computer Engineering, University of Victoria, P.O. Box 3055, STN CSC, Victoria, B.C., Canada V8W 3P6 agullive@engr.uvic.ca Abstract. This paper studies senary simplex codes of type α and two punctured versions of these codes (type β and γ). Self-orthogonality, torsion codes, weight distribution and weight hierarchy properties are studied. We give a new construction of senary codes via their binary and ternary counterparts, and show that type α and β simplex codes can be constructed by this method. 1 Introduction There has been much interest in codes over finite rings in recent years, especially the rings ZZ 2k where ZZ 2k denotes the ring of integers modulo 2k. In particular codes over ZZ 4 have been widely studied [1], [5],[6],[7],[8], [9],[10],[11], [12]. More recently ZZ 4 -simplex codes (and their Gray images), have been investigated by Bhandari, Gupta and Lal in [2]. Good binary linear and non-linear codes can be obtained from codes over ZZ 4 via the Gray map. Thus it is natural to investigate simplex codes over the ring ZZ 2k . In particular, one can construct mixed binary/ternary codes via senary codes by applying the Chinese Gray map (see Example 1). Motivated by this (apart from practical applications such as PSK modulation [4]), in this paper we consider senary simplex codes, and investigate their fundamental properties. We also study their Chinese product type construction. A linear code C, of length n, over ZZ 6 is an additive subgroup of ZZ n6 . An element of C is called a codeword of C and a generator matrix of C is a matrix whose rows generate C. The Hamming weight wH (x) of a vector x in ZZ n6 is the number of non-zero Pn components. The Lee weight wL (x) of a vector x = (x1 , x2 , . . . ,P xn ) is i=1 min{|xi |, |6 − xi |}. The Euclidean weight wE (x) of a n vector x is i=1 min{x2i , (6 − xi )2 }. The Euclidean weight is useful in connection with latticePconstructions. The Chinese Euclidean weight wCE (x) of a vector 
i x ∈ ZZ nm is ni=1 2 − 2 cos 2πx . This is useful for m−P SK coding [4]. The m Hamming, Lee and Euclidean distances dH (x, y), dL (x, y) and dE (x, y) between two vectors x and y are wH (x − y), wL (x − y) and wE (x − y), respectively. The minimum Hamming, Lee and Euclidean weights, dH , dL and dE , of C are the 2 Manish K. Gupta et al. smallest Hamming, Lee and Euclidean weights among all non-zero codewords of C, respectively. The Chinese Gray map φ : ZZ n6 → ZZ n2 ZZ n3 is the coordinate-wise extension of the function from ZZ 6 to ZZ 2 ZZ 3 defined by 0 → (0, 0), 1 → (1, 1), 2 → (0, 2), 3 → (1, 0), 4 → (0, 1) and 5 → (1, 2). The inverse map φ−1 is a ring isomorphism and so is φ[6]. The image φ(C), of a linear code C over ZZ 6 of length n by the Chinese Gray map, is a mixed binary/ternary code of length 2n. The dual code C ⊥ of C is defined as {x ∈ ZZ n6 | x · y = 0 for all y ∈ C} where x · y is the standard inner product of x and y. C is self-orthogonal if C ⊆ C ⊥ and C is self-dual if C = C ⊥ . Two codes are said to be equivalent if one can be obtained from the other by permuting the coordinates and (if necessary) changing the signs of certain coordinates. Codes differing by only a permutation of coordinates are called permutation-equivalent. In this paper we define ZZ 6 -simplex codes of type α, β and γ namely, Skα , β Sk and Skγ , and determine some of their fundamental parameters. Section 2 contains some preliminaries and notations. Definitions and basic parameters of ZZ 6 -simplex codes of type α, β and γ are given in Section 3. Section 4 investigates their Chinese product type construction. 2 Preliminaries and Notations Any linear code C over ZZ 6 is permutation-equivalent to a code with generator matrix G (the rows of G generate C) of the form   Ik1 A1,2 A1,3 A1,4 G =  0 2Ik2 2A2,3 2A2,4  , (1) 0 0 3Ik3 3A3,4 where the Ai,j are matrices with entries 0 or 1 for i > 1, and Ik is the identity matrix of order k. Such a code is said to have rank {1k1 , 2k2 , 3k3 } or simply rank {k1 , k2 , k3 } and |C| = 6k1 3k2 2k3 [1]. If k2 = k3 = 0 then the rank of C is {k1 , 0, 0} or simply k1 = k. To each code C one can associate two residue codes viz C2 and C3 defined as C2 = {v | v ≡ w (mod 2), w ∈ C}, C3 = {v | v ≡ w (mod 3), w ∈ C}. and Code C2 is permutation-equivalent to a code with generator matrix of the form   Ik1 A1,2 A1,3 A1,4 , (2) 0 0 3Ik3 3A3,4 where Ai,j are binary matrices for i > 1. Note that C2 has dimension k1 + k3 . The ternary code C3 is permutation-equivalent to a code with generator matrix Lecture Notes in Computer Science of the form  Ik1 A1,2 A1,3 A1,4 0 2Ik2 2A2,3 2A2,4  , 3 (3) where Ai,j are binary matrices for i > 1. Note that C3 has dimension k1 + k2 . One can also associate two torsion codes with C viz C2 ⋆ and C3⋆ defined as o nc | c = (c1 , . . . , cn ) ∈ C and ci ≡ 0 (mod 3) for 1 ≤ i ≤ n C2⋆ = 3 and C3⋆ = nc 2 | c = (c1 , . . . , cn ) ∈ C and ci ≡ 0 o (mod 2) for 1 ≤ i ≤ n . If k2 = k3 = 0 then Ci = Ci⋆ for i = 2, 3. A linear code C over ZZ 6 of length n and rank {k1 , k2 , k3 } is called an [n; k1 , k2 , k3 ] code. If k2 = k3 = 0, C is called an [n, k] code. In the case of simplex codes we indeed have k2 = k3 = 0. Let C : [n; k1 , k2 , k3 ] be a code over ZZ 6 . For r1 ≤ k1 , r2 ≤ k2 , r1 + r2 + r3 ≤ k1 + k2 + k3 , the Generalized Hamming Weight of C is defined by dr1 ,r2 ,r3 = min {wS (D) | D is an [n; r1 , r2 , r3 ] 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 {dr1 ,r2 ,r3 } is called the weight hierarchy of C. We have the following Lemma connecting the support weight and the Chinese Euclidean weight. Lemma 1. Let D : [n; r1 , r2 , r3 ] be a senary linear code then X wCE (c) = 2r1 +r3 +1 · 3r1 +r2 · wS (D). c∈D Proof. Consider the (r×n) array of all the codewords in D (where r = 6r1 3r2 2r3 ). It is easy to see that each column consists of either 1. 2. 3. 4. only zeros 0 and 3 equally often 0, 2 and 4 equally often 0, 1, 2, 3, 4, 5 equally often. Let ni , i = 0, 1, 2, 3 be the number of columns of each type. Then n1 + n2 + n3 = wS (D). Now applying the standard arguments to evaluate the sum yields the result. Thus for any linear code C over ZZ 6 , dr1 ,r2 ,r3 may also be defined by ( ) X 1 dr1 ,r2 ,r3 = r1 +r3 +1 r1 +r2 min wCE (c) | D is an [n; r1 , r2 , r3 ] subcode of C . 2 ·3 c∈D 4 3 Manish K. Gupta et al. Senary Simplex Codes of Type α, β and γ k k Let Gα k be a k × 2 3 matrix over ZZ 6 consisting of all possible distinct columns. α Inductively, Gk may be written as   00 · · · 0 11 · · · 1 22 · · · 2 33 · · · 3 44 · · · 4 55 · · · 5 α Gk = α α α α α Gα k−1 Gk−1 Gk−1 Gk−1 Gk−1 Gk−1 k×6k α α k with Gα 1 =[012345]. The code Sk generated by Gk has length 6 and the rank α of Sk is {k, 0, 0}. The following observations are useful to obtain the weight distribution of Skα . Remark 1. If Ak−1 denotes the (6k−1 × 6k−1 ) array consisting of all codewords α in Sk−1 , and if J is the matrix of all 1′ s then the (6k × 6k ) array of codewords of Skα is given by   Ak−1 Ak−1 Ak−1 Ak−1 Ak−1 Ak−1  Ak−1 J + Ak−1 2J + Ak−1 3J + Ak−1 4J + Ak−1 5J + Ak−1     Ak−1 2J + Ak−1 4J + Ak−1 Ak−1 2J + Ak−1 4J + Ak−1   .  Ak−1 3J + Ak−1 Ak−1 3J + Ak−1 Ak−1 3J + Ak−1     Ak−1 4J + Ak−1 2J + Ak−1 Ak−1 4J + Ak−1 2J + Ak−1  Ak−1 5J + Ak−1 4J + Ak−1 3J + Ak−1 2J + Ak−1 1J + Ak−1 Remark 2. If R1 , R2 , ..., Rk denote the rows of the matrix Gα k then wH (Ri ) = 5 · 6k−1 , wL (Ri ) = 9 · 6k−1 , wE (Ri ) = 19 · 6k−1 and wCE (Ri ) = 2 · 6k . It may be observed that each element of ZZ 6 occurs equally often in every row of Gα k . Let c = (c1 , c2 , . . . , cn ) ∈ C. For each j ∈ ZZ 6 let ωj (c) = |{i | ci = j}|. We have the following lemma. Lemma 2. Let c ∈ Skα , c 6= 0. 1. If for at least one i, ci is a unit (1 or 5) then ∀j ∈ ZZ 6 ωj = 2k−1 · 3k−1 in c. 2. If ∀i, ci ∈ {0, ±2} then ∀j ∈ {0, ±2} ωj = 2k · 3k−1 in c. 3. If ∀i, ci ∈ {0, 3} then ∀j ∈ {0, 3} ωj = 2k−1 · 3k in c. α Proof. By Remark 1, any x ∈ Sk−1 gives rise to six codewords of Skα :
y1 = x x x x x x ,
y2 = x 1 + x 2 + x 3 + x 4 + x 5
+ x , y3 = x 2 + x 4 + x x 2 + x 4
+ x , y4 = x 3 + x x 3 + x x 3 + x ,
y5 = x 4 + x 2 + x x 4 + x 2 + x , and
y6 = x 5 + x 4 + x 3 + x 2 + x 1 + x , where i = (iii...i). Now the result can be easily proved by induction on k. Lecture Notes in Computer Science 5 Now we recall some known facts about binary and ternary simplex codes. Let G(Sk ) (columns consisting of all non-zero binary k-tuples) be a generator matrix for an [n, k] binary simplex code Sk . Then the extended binary simplex code (also known as a type α binary simplex code), Sˆk is generated by the matrix G(Sˆk ) = [0 G(Sk )]. Inductively,   00 . . . 0 11 . . . 1 ˆ ˆ (4) G(Sk ) = ˆ ) G(Sk−1 ˆ ) with G(S1 ) = [01] . G(Sk−1 The ternary simplex code of type α is defined inductively by   00 · · · 0 11 · · · 1 22 · · · 2 with T1α = [012] , Tkα = α α α Tk−1 Tk−1 Tk−1 (5) and the ternary simplex code is defined by the usual generator matrix as     11 · · · 1 00 · · · 0 111 0 β Tkβ = with T = . β 2 α 012 1 Tk−1 Tk−1 Now we determine the torsion codes of the senary simplex code of type α. Lemma 3. The binary (ternary) torsion code of Skα is equivalent to 3k copies
of the binary type α simplex code 2k copies of the ternary type α simplex code . Proof. We will prove the binary case by induction on k. The proof of ternary case is similar and so is omitted. Observe that the binary torsion code of Skα is the set of codewords obtained by replacing 3 by 1 in all 2-linear combinations of the rows of the matrix   00 · · · 0 33 · · · 3 00 · · · 0 33 · · · 3 00 · · · 0 33 · · · 3 α 3Gk = . (6) α α α α α 3Gα k−1 3Gk−1 3Gk−1 3Gk−1 3Gk−1 3Gk−1 Clearly the result holds for k = 2. Assuming that the binary torsion code is equivalent to the 3k−1 copies of the extended binary simplex code, we have  ˆ ) in place of 3Gα ˆ ) · · · 3G(Sk−1 3G(Sk−1 k−1 in the above matrix. Now regrouping the columns in the above matrix according to (4) yields the desired result. As a consequence of Lemmas 2 and 3, one gets the weight distribution of Skα . Theorem 1. The Hamming, Lee, Euclidean and C-Euclidean weight distributions of Skα are 1. AH (0) = 1, AH (3 · 6k−1 ) = (2k − 1), AH (4 · 6k−1 ) = (3k − 1), AH (5 · 6k−1 ) = (2k − 1)(3k − 1). 2. AL (0) = 1, AL (8 · 6k−1 ) = (3k − 1), AL (9 · 6k−1 ) = 3k (2k − 1) − 1. 3. AE (0) = 1, AE (27 · 6k−1 ) = (2k − 1), AE (16 · 6k−1 ) = (3k − 1), AE (19 · 6k−1 ) = (2k − 1)(3k − 1). 4. ACE (0) = 1, ACE (2 · 6k ) = 3k · 2k − 1, where AH (i) (AL (i)) denotes the number of vectors of Hamming (Lee) weight i in Skα , and similarly for the Euclidean weights of both types. 6 Manish K. Gupta et al. Proof. By Lemma 2, each non-zero codeword of Skα has Hamming weight either 3 · 6k−1 , 4 · 6k−1 , or 5 · 6k−1 and Lee weight either 8 · 6k−1 or 9 · 6k−1 . Since the dimension of the binary torsion code is k, there will be 2k − 1 codewords of the Hamming weight 3 · 6k−1 , and the dimension of the ternary torsion code is k, so there will be 3k − 1 codewords of the Hamming weight 4 · 6k−1 . Hence the
number of codewords having Hamming weight 5·6k−1 will be 6k − 3k + 2k − 1 . Similar arguments hold for the other weights. The symmetrized weight enumerator (swe) of a senary code C is defined as X sweC (a, b, c, d) := an0 (x) bn1 (x) cn2 (x) dn3 (x) , x∈C where ni (x) denotes the number of j such that xj = ±i. Let S¯kα be the punctured code of Skα obtained by deleting the zero coordinate. Then the swe of S¯kα is k−1 sweS¯kα (a, b, c, d) = 1 + (2k − 1)d(ad)3·6 k (3 − 1)a k−1 2·6 −1 4·6 c −1 k−1 + k−1 + (2k − 1)(3k − 1)d(ad)6 −1 k−1 (bc)2·6 Remark 3. 1. Skα is an equidistant code with respect to Chinese Euclidean distance whereas the binary (quaternary i.e, over ZZ 4 ) simplex code is equidistant with respect to Hamming (Lee) distance. 2. The minimum weights of Skα are: dH = 3 · 6k−1 , dL = 8 · 6k−1 , dE = 16 · 6k−1 , dCE = 2 · 6k . Example 1. Consider the 64 = 1296 codewords of the senary code generated by the following generator matrix 11111111 22220000 22002200 20202020 . 33334444 33443344 34343434 Using the Chinese Gray map results in a mixed code with 8 binary and 8 ternary coordinates, which gives N (8, 8, 4) ≥ 1296, while the ternary code of length 8, dimension 4 and distance 4 is optimal [3]. Let Λk be the k × 3k · (2k − 1) matrix defined inductively by Λ1 = [135] and   0...0 1···1 2···2 3···3 4···4 5···5 Λk = , α α Λk−1 Gα k−1 Λk−1 Gk−1 Λk−1 Gk−1 for k ≥ 2; and let µk be the k × 2k−1 · (3k − 1) matrix defined inductively by µ1 = [12] and   0···0 1···1 2···2 3···3 µk = , α µk−1 Gα k−1 Gk−1 µk−1 . Lecture Notes in Computer Science 7 α for k ≥ 2, where Gα k−1 is the generator matrix of Sk−1 . Now let Gβk be the k × (2k −1)(3k −1) 2 matrix defined inductively by  111111 0 222 33 β , G2 = 012345 1 135 12  and for k > 2 Gβk =   11 · · · 1 00 · · · 0 22 · · · 2 33 · · · 3 , β Gα Λk−1 µk−1 k−1 Gk−1 β α α where Gα k−1 is the generator matrix of Sk−1 . Note that Gk is obtained from Gk by k k k+1 columns. By induction it is easy to verify that no two deleting (2 +1)(3 2−1)+2 β columns of Gk are hmultiples of each other. Let Skβ be the code generated by Gβk . i k k −1) , k code. To determine the weight distributions Note that Skβ is a (2 −1)(3 2 of Skβ we first make some observations. The proof of the following proposition is similar to that of Proposition 2. Proposition 1. Each row of Gβk contains 6k−1 units and ω2 + ω4 = 3k−1 (2k−1 − 1), ω3 = 2k−2 (3k−1 − 1), ω0 = (2k−1 −1)(3k−1 −1) . 2
Remark 4. Each row of Gβk has Hamming weight 3k−1
· (2k − 1) + 2k−2 · (3k−1 − 1) , Lee weight 2 · 3k−1 (3 · 2k−2 −
1) + 3 · 2k−2 (3k−1 − 1) , Euclidean weight 3k−1 (19 · 2k−2 − 4) − 9 · 2k−2 , and Chinese Euclidean weight 6k − 2k − 3k . The proof of the following lemma is similar to the proof of Lemma 2. Lemma 4. Let c ∈ Skβ , c 6= 0. 1. If for at least one i, ci is a unit then ∀j ∈ ZZ 6 ω1 + ω5 = 6k−1 , k−1 k−1 −1) in c. ω2 + ω4 = 3k−1 (2k−1 − 1), ω3 = 2k−2 (3k−1 − 1), ω0 = (2 −1)(3 2 k k−1 2. If ∀i, ci ∈ {0, ±2} then ∀j ∈ {0, ±2} ω2 +ω4 = 3k−1 (2k −1), ω0 = (2 −1)(32 −1) in c. k−1 k −1) 3. If ∀i, ci ∈ {0, 3} then ∀j ∈ {0, 3} ω3 = 2k−2 (3k − 1), ω0 = (2 −1)(3 in 2 c. The proof of the following lemma is similar to that of Lemma 3 and is omitted. k Lemma 5. The binary (ternary) torsion code of Skβ is equivalent to (3 2−1)
copies of the binary simplex code (2k − 1) copies of the ternary simplex code . The proof of the following theorem is similar to that of Theorem 1 and is omitted. Theorem 2. The Hamming, Lee weight, Euclidean and C-Euclidean weight distributions of Skβ are:

1), AH 3k−1 · (2k − 1) = (3k − 1), 1. AH (0) = 1, AH 2k−2 · (3k − 1) = (2k −
AH 3k−1 · (2k − 1) + 2k−2 · (3k−1 − 1) = (2k − 1)(3k − 1). 8 Manish K. Gupta et al.

2. AL (0) = 1, AL 3 · 2k−2 (3k − 1) = (2k −1), AL
2 · 3k−1 (2k − 1) = (3k −1), AL 2 · 3k−1 (3 · 2k−2 − 1) + 3 · 2k−2 (3k−1 − 1) = (2k − 1)(3k − 1).

k − 1), AE 4 · 3k−1 (2k − 1) = (3k − 3. AE (0) = 1, AE 9 · 2k−2 (3k − 1) = (2
1), AE 3k−1 (19 · 2k−2 − 4) − 9 · 2k−2 = (2k − 1)(3k − 1). 4. ACE (0) = 1, ACE (6k − 2k ) = (2k − 1), ACE (6k − 3k ) = (3k − 1), ACE (6k − 2k − 3k ) = (2k − 1)(3k − 1), where AH (i) (AL (i)) denotes the number of vectors of Hamming (Lee) weight i in Skα , and similarly for the Euclidean weights of both types. Remark 5. 1. The swe of Skβ is given as swe(a, b, c, d) = 1 + 3−k p(k)an(k−1)+p(k−1) d2 n 2 −k+1 k q(k)a n(k−1) a −1 q(k) + −1 p(k) + 3−k p(k)b6 q(k−1) 3 c k k−1 o cp(k−1) dq(k−1) . −1) , p(k) = 3k (2k − 1) and q(k) = 2k−1 (3k − 1). where n(k) = (2 −1)(3 2 2. The minimum weights of Skβ are: dH = 2k−2 (3k − 1), dL = 2 · 3k−1 (2k − 1), dE = 4 · 3k−1 (2k − 1), dCE = 6k − 2k − 3k . Let Gγk be the k × 2k−1 (3k − 2k ) matrix defined inductively by Gγ2 =   111111 0 2 3 4 , 012345 1 1 1 1 and for k > 2 Gγk =   11 · · · 1 00 · · · 0 22 · · · 2 33 · · · 3 44 · · · 4 , γ γ γ γ Gα k−1 Gk−1 Gk−1 Gk−1 Gk−1 γ α α where Gα k−1 is the generator matrix of Sk−1 . Note that Gk is obtained from Gk k−1 k k by deleting 2 (2 + 3 ) columns. By induction it is easy to verify that no two columns of Gγk aremultiples of each other. Let Skγ be the code generated by Gγk . Note that Skγ is a 2k−1 (3k − 2k ), k code. Proposition 2. Each row of Gγk contains 6(k−1) units and ω0 = ω2 = ω3 = ω4 = 2k−2 (3k−1 − 2k−1 ). Proof. Clearly the assertion holds for the first row. Assume that the result holds for each row of Gγk−1 . Then the number of units in each row of Gγk−1 is 6(k−2) . By k−1 Lemma 2, the number of units in any row of Gα · 3k−2 . Hence the total k−1 is 2 γ k−1 k−2 k−2 k−2 number of units in any row of Gk will be 2 ·3 + 4 ·2 ·3 = 2k−1 ·3k−1 . A similar argument holds for the number of 0’s, 2’s, 3’s and 4’s.   Remark 6. Each row of Gγk has Hamming weight 3 · 2k−2 5 · 3k−2 − 2k−1 , Lee  weight 2k−2 3k+1 − 7 · 2k−1 , Euclidean weight 2k−2 19 · 3k−1 − 17 · 2k−1 , and Chinese Euclidean weight 6k − 5 · 4k−1 . Lecture Notes in Computer Science 9 The various weight distributions of Skγ can be obtained using arguments similar to other simplex codes. To save the space we omit them. The weight hierarchy of Skα is given by the following theorem. Theorem 3. The weight hierarchy of Skα is given by dr1 ,r2 ,r3 (Skα ) = 6k − 3k−r1 −r2 · 2k−r1 −r3 . Proof. By Remark 3 and the definition of dr1 ,r2 ,r3 after the Lemma 1. 4 Chinese Product Type Construction The Chinese remainder theorem (CRT) plays an important role in the study of codes over ZZ 2k [4, 6]. In particular, given binary and ternary linear codes of length n and dimension k, one can construct a senary code (over ZZ 6 ) of length n using CRT. The following theorem is from [4, 6]. Theorem 4. [4, 6] If B and T are linear codes  of length n over GF (2) and GF (3), respectively, then the set CRT (B, T ) = φ−1 (cb , ct ) | cb ∈ B, ct ∈ T is a linear code of length n over ZZ 6 . Moreover if B and T are self-orthogonal then CRT (B, T ) is also self-orthogonal. If generator matrices of B, T and CRT (B, T ) are G(B), G(T ) and G(CRT (B, T )), respectively, then we have φ(G(CRT (B, T ))) = [G(B)|G(T )] , where φ is the Chinese Gray map. If the codes B and T are of different lengths, say, n1 and n2 then it seems that no non-trivial method is known to construct a code over ZZ 6 from these codes. In the trivial case of course one can add extra zero columns to the generator matrix of the code of shorter length and then use Theorem 4. Here we present a new construction of a generator matrix of senary code from codes of different lengths. Let G(B) = [x1 x2 . . . xn1 ] and G(T ) = [y1 y2 . . . yn2 ] where xi , yi are the corresponding columns. Now form the matrix G(B)⋆G(T ) consisting of the n1 n2 1 pairs of total 2n1 n2 columns {xi y1 xi y2 . . . xi yn2 }ni=1 . These pairs of columns give a generator matrix of length n1 n2 (the product of the lengths of the binary and ternary codes) over ZZ 6 using the inverse Chinese Gray map. In particular, if n1 = n2 = n then we get a code of length n2 . Note that if we use the Theorem 4 to construct a generator matrix for the case of n1 = n2 = n, we obtain a code of length n with generator matrix [x1 y1 x2 y2 . . . xn yn ]. In this case, the resulting code will be self orthogonal if the corresponding binary and ternary codes are self orthogonal [6]. Similarly it is easy to see that Lemma 6. The senary codes constructed by G(B) ∗ G(T ) will be self orthogonal if the corresponding codes B and T are self orthogonal. The next two results show that self-orthogonal simplex codes of type α and β can be obtained from the construction G(B) ∗ G(T ). 10 Manish K. Gupta et al. Theorem 5. The codes Skα and Skβ can be obtained via the construction G(B) ∗ G(T ). Proof. We will only prove the result for Skα , since the other case is similar. If we apply the Chinese Gray map to the generator matrix Gα k , we see that it is equivalent to the matrix G(Sˆk ) ⋆ Tkα , where Tkα is defined in (5). Theorem 6. The codes Skα (k ≥ 3) and Skβ (k ≥ 2) are self orthogonal. Proof. The result follows from Lemma 6 and Theorem 5. It can also be proved by induction on k since the rows of the generator matrices are pairwise orthogonal and each of the rows has Euclidean weight a multiple of 12 [1]. Remark 7. The code Skγ is not self-orthogonal as the Euclidean weights of the rows of Gγk are not a multiple of 12. Acknowledgement. The authors would like to thank Patrick Solé for providing copies of [6] and [12], and also Patric R.J. Östergård for providing a copy of [13]. References 1. Bannai E., Dougherty S.T., Harada M. and Oura M., Type II codes, even unimodular lattices and invariant rings. 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