Additive Codes over Z2 × Z4 J. Borges, S.T. Dougherty, C. Fern´andez-C´ordoba June 22, 2011 Abstract We describe recent results for codes over Z2 ×Z4 giving their connection to binary codes via a natural Gray map. We study Z2 Z4 self-dual codes and we state the major results concerning these codes. We state several open questions and discuss possible avenues of research.

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Introduction

β The codes we will consider are subgroups of the ambient space Zα 2 × Z4 , where Z2 and Z4 are the rings of integers modulo 2 and modulo 4, respectively. In Delsarte’s landmark paper in 1973 [7], he defines additive codes as subgroups of the underlying abelian group in a translation association scheme. For the binary Hamming scheme, namely when the underlying abelian group is of order β 2n , the only structures are those of the form Zα 2 × Z4 , with α + 2β = n ([8]). β Therefore, the subgroups C of Zα 2 × Z4 are the only additive codes in the binary Hamming scheme. We shall refer to them as Z2 Z4 codes, and we will assume they are linear (i.e. subgroups unless otherwise specified). See the following for more information [1, 2, 3, 4, 5, 6, 9, 10, 11]. β We adopt the following notation. If C ⊆ Zα 2 × Z4 is a Z2 Z4 code and v ∈ C, β then v = (x, y) where x = (x1 , . . . , xα ) ∈ Zα 2 and y = (y1 , . . . , yβ ) ∈ Z4 . The impetus for the study of codes over rings is often marked by the understanding of the Gray map from Z4 to Z2 and the proof that certain well known binary codes were actually the image under this map of linear Z4 codes. We can extend the definition of the usual Gray map to the space described β n above as follows: Φ : Zα 2 × Z4 −→ Z2 , where n = α + 2β, given by Φ(x, y) = (x, φ(y1 ), . . . , φ(yβ )), where φ : Z4 −→ Z22 is the usual Gray map, namely φ(0) = (0, 0), φ(1) = (0, 1), φ(2) = (1, 1), φ(3) = (1, 0). This map is an isomβ etry which transforms Lee distances defined in Zα 2 × Z4 to Hamming distances α+2β defined in Z2 . We shall make this explicit by defining the weight of a Z2 Z4 vector. For β vectors v1 ∈ Zα 2 and v2 ∈ Z4 , we let wtH (v1 ) be the Hamming weight of v1 β and wtL (v2 ) be the Lee weight of v2 . For a vector v = (v1 , v2 ) ∈ Zα 2 × Z4 , we define the weight of v, denoted by wt(v), as wtH (v1 ) + wtL (v2 ), or equivalently, the Hamming weight of Φ(v). β γ δ The code C is a subgroup of Zα 2 × Z4 , as such it is isomorphic to Z2 × Z4 for γ δ some γ and δ. We say that C is of type 2 4 as a group. It follows that it has |C| = 2γ+2δ codewords and the number of order two codewords in C is 2γ+δ .

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Denote by X and Y the set of Z2 and Z4 coordinate positions respectively. It follows immediately that |X| = α and |Y | = β. The codes CX and CY are the punctured codes of C formed by deleting the coordinates outside of X and Y respectively. Denote by Cb the subcode of C which contains all order two codewords and let κ be the dimension of the binary linear code (Cb )X . We then say that C is of type (α, β; γ, δ; κ). Although a Z2 Z4 -additive code C is not a free module, every codeword is uniquely expressible in the form γ X

λi ui +

i=1

δ X

µj vj ,

j=1

where λi ∈ Z2 for 1 ≤ i ≤ γ, µj ∈ Z4 for 1 ≤ j ≤ δ and ui , vj are vectors β in Zα 2 × Z4 of order two and four, respectively. The vectors ui , vj give us a generator matrix G of size (γ + δ) × (α + β) for the code C. We can write G as   B1 2B3 , (1) G= Q B2 where B1 , B2 , B3 are matrices over Z2 of size γ ×α, δ ×α and γ ×β, respectively; and Q is a matrix over Z4 of size δ × β with quaternary row vectors of order four. Moreover, we can get a canonical generator matrix, which we describe in the following theorem. Let Ik be the identity matrix of size k × k. Theorem 1 ([5]) Let C be a Z2 Z4 -additive code of type (α, β; γ, δ; κ). Then, C is permutation equivalent to a Z2 Z4 -additive code with canonical generator matrix of the form   Iκ Tb 2T2 0 0 GS =  0 0 2T1 2Iγ−κ 0  , (2) 0 Sb Sq R Iδ where Tb , Sb are matrices over Z2 ; T1 , T2 , R are matrices over Z4 with all entries in {0, 1} ⊂ Z4 ; and Sq is a matrix over Z4 . The generator for the orthogonal code also has a general form as given in the following theorem. Theorem 2 ([5]) Let C be a Z2 Z4 -additive code of type (α, β; γ, δ; κ) with canonical generator matrix (2). Then, the generator matrix of C ⊥ is   t 0 0 2Sbt Tb Iα−κ , 0 0 2Iγ−κ 2Rt (3) HS =  0
t t t 0 Iβ+κ−γ−δ T1 − Sq + RT1 T2 where Tb , T2 are matrices over Z2 ; T1 , R, Sb are matrices over Z4 with all entries in {0, 1} ⊂ Z4 ; and Sq is a matrix over Z4 . Moreover, T2 and Sb are obtained from the matrices of (2) with the same name, whose all entries are {0, 1}, after applying the modulo 2 map and the inclusion, respectively.

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In general, the binary image under the Gray map of a Z2 Z4 -additive code, C is not linear. When C = φ(C) is not linear, then we define invariants for Z2 Z4 linear codes, the rank r and dimension of the kernel k. The rank of C is simply the dimension of the linear span of the codewords of C. The kernel of C is the set of vectors that leave C invariant under translation, i.e. {x ∈ Zn2 | C + x = C}. If C contains the all-zero vector, then the kernel of C is a binary linear subcode of C. In [9] the possible values of r, k and also the possible pair (r, k) are established. Finally, a construction of a code with each possible r, k and pair (r, k) is given. The duality for Z2 Z4 -additive codes was given in [5]. Namely, for any two β vectors u, v ∈ Zα 2 × Z4 we define α+β α X X hu, vi = 2( uj vj ∈ Z4 . ui vi ) + j=α+1

i=1

Then, the additive dual code of C, denoted by C ⊥ , is defined in the standard way, that is: β C ⊥ = {v ∈ Zα 2 × Z4 | hu, vi = 0 for all u ∈ C}.

The code C is called self-orthogonal if C ⊥ ⊆ C and self-dual if C ⊥ = C. If C = φ(C), the binary code Φ(C ⊥ ) is denoted by C⊥ and called the Z2 Z4 -dual code of C. In [5] it was proved that the additive dual code C ⊥ , which is also a ¯κ Z2 Z4 -additive code, is of type (α, β; γ¯ , δ; ¯ ), where γ¯ = α + γ − 2κ, δ¯ = β − γ − δ + κ, κ ¯ = α − κ.

(4)

The importance of these codes is immediate after the following theorem. Theorem 3 ([4, 8, 11]) Let C be a Z2 Z4 -additive code of type (α, β; γ, δ; κ) and C ⊥ its additive dual code. Then, |C||C ⊥ | = 2n , where n = α + 2β. Define the usual Hamming weight enumerator of C to be X WC (x, y) = xn−wt(c) y wt(c) , c∈C

where n = α + 2β. We have that WC ⊥ (x, y) =

1 WC (x + y, x − y). |C|

The codes C and C ⊥ can have non-linear images whose weight enumerators are dual with respect to the MacWilliams relations. That is, they are similar to linear codes but are not linear themselves. We can now state our first open problem. Open Problem 1 Find optimal Z2 Z4 codes whose images have minimum distances that are higher than any binary linear code. Optimal linear Z2 Z4 codes are studied in [1]. 3

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Z2 Z4 -additive self-dual codes

Given the results in the first section it is natural to study Z2 Z4 self-dual codes. These codes were studied in [2]. Their importance lies in the fact that the image under the Gray map of a code C is a (possibly) non-linear code whose weight enumerator is the weight enumerator of a formally self-dual code. Let C be a Z2 Z4 -additive code. If C = CX × CY , then C is called separable, otherwise it is called non-separable. If a Z2 Z4 -additive self-dual code has odd weights, then we say it is Type 0. If it has only even weights, then we say the code is Type I. If all the codewords have doubly-even weight we say the code is Type II. These definitions are made this way so that the image of the code under the Gray map is a binary code with the weight enumerator of the same Type (except Type 0 which is usually not defined in the binary case.) We say that a binary code C is antipodal if for any codeword z ∈ C, then z + 1 ∈ C. If C is a Z2 Z4 -additive code, we say that C is antipodal if Φ(C) is antipodal. The standard invariant theory for weight enumerators, shadow theory, and existence theorems are examined for these self-dual codes. The fundamental results of [2] are summarized as follows:

separable/ non-separable antipodality WC (x, y) separable α, β; a, b ≥ 0 non-separable α, β; a, b ≥ 0

Type 0 non-separable non-antipodal

Type I separable or non-separable antipodal

Type II separable or non-separable antipodal

C[x2 + y 2 , y(x − y)]

C[x2 + y 2 , x y (x2 − y 2 )2 ]

C[x8 + 14x4 y 4 + y 8 , x4 y 4 (x4 − y 4 )4 ]

α = 2 + 2a β =2+b

α = 2 + 2a β =1+b α = 4 + 2a β =4+b

α = 8 + 8a β = 4 + 4b α = 8 + 8a β = 4 + 4b

2 2

In [2], there are also contructions for codes of Type 0, I and II, both separable and non-separable, for all possible values of α and β. Open Problem 2 Find Z2 Z4 self-dual codes whose weight enumerator is the weight enumerator of a putative binary self-dual code. Find Z2 Z4 self-dual codes with a given weight enumerator where it is known that no binary self-dual code exists with that weight enumerator. In [2], a generalization of the neighbor construction is given. In [3], generalizations of the building up construction or shadow construction are given. Specifically, for a code C we take a vector v ∈ / C and define Cv to be the subcode of C of vectors that are orthogonal to C. Then we adjoin coordinates to the vectors in Cv⊥ to make this extended code self-dual. Constructions are given for extended the binary part, the quaternary part and both parts simultaneously. Examples of codes from all of these constructions can be found in [3]. Open Problem 3 Generalize other constructions of self-dual codes over fields and rings to Z2 Z4 codes.

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References [1] M. Bilal, J. Borges, S.T. Dougherty, C. Fern´andez-C´ordoba, “Optimal Codes over Z2 × Z4 ”, in preparation. [2] J. Borges, S.T. Dougherty, C. Fern´andez-C´ordoba, “Self-Dual Codes over Z2 × Z4 ”, in submission. [3] J. Borges, S.T. Dougherty, C. Fern´andez-C´ordoba, “Constructions of SelfDual Codes over Z2 × Z4 ”, in preparation. [4] J. Borges, C. Fern´ andez-C´ordoba, J. Pujol, J. Rif`a and M. Villanueva, “On Z2 Z4 -linear codes and duality”, V Jornades de Matem` atica Discreta i Algor´ısmica, Soria (Spain), Jul. 11-14, pp. 171-177, 2006. [5] J. Borges, C. Fern´ andez-C´ordoba, J. Pujol, J. Rif`a and M. Villanueva, “Z2 Z4 -linear codes: generator matrices and duality”, Designs, Codes and Cryptography, vol. 54(2), pp. 167-179, 2010. [6] J. Borges and J. Rif` a, “A characterization of 1-perfect additive codes”, IEEE Trans. Inform. Theory, vol. 45(5), pp. 1688-1697, 1999. [7] P. Delsarte, “An algebraic approach to the association schemes of coding theory”, Philips Res. Rep. Suppl., vol. 10, 1973. [8] P. Delsarte, V. Levenshtein, “Association Schemes and Coding Theory”, IEEE Trans. Inform. Theory, vol. 44(6), pp. 2477-2504, 1998. [9] C. Fern´ andez-C´ ordoba, J. Pujol and M. Villanueva, “Z2 Z4 -linear codes: rank and kernel”, to appear in Designs, Codes and Cryptography, 2010. [10] K.T. Phelps and J. Rif` a, “On binary 1-perfect additive codes: some structural properties”, IEEE Trans. Inform. Theory, vol. 48(9), pp. 2587-2592, 2002. [11] J. Pujol and J. Rif` a, “Translation invariant propelinear codes”, IEEE Trans. Inform. Theory, vol. 43, pp. 590-598, 1997.

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