On Z2 Z4 -linear codes and duality* Borges, J.; Fernández, C.; Pujol, J.; Rifà, J.; Villanueva, M. Department of Information and Communications Engineering Universitat Autònoma de Barcelona. 08193-Bellaterra, Spain codis@deic.uab.cat Abstract. A binary Z2 Z4 -linear code is basically a combination of a linear code and a Z4 -linear code. Changing ones by twos in the Z2 part, one can think that this is equivalent to a Z4 -linear code. However, by considering duality, we show that the situation is not equivalent. We will compute the parameters of dual codes, generator and parity check matrices for these Z2 Z4 -linear codes. Keywords. Z2 Z4 -linear, duality, generator matrix, parity check matrix. 1. Introduction and basic definitions Let Z2 and Z4 be the ring of integers mod 2 and mod 4, respectively. Let F be the set of all binary vectors of length n and let Zn4 be the set of all quaternary vectors or vectors over Z4 of length n. Any non-empty subset C of F is a binary code and a subgrup of F is called a binary linear code or a Z2 -linear code. Equivalently, any non-empty subset C of Zn4 is a quaternary code and a subgrup of Zn4 is called a quaternary linear code. Binary linear codes have been generalized in different ways. We usually consider a binary code C with an operation ⋆ (not necessarily the standard sum over F ) such that (C, ⋆) is a group. As usual, we also consider the Hamming metric, hence we want that the operation ⋆ is Hamming-compatible [2], in the sense that d(x, x ⋆ y) = w(y), where d(·, ·) and w(·) denote the Hamming distance and the Hamming weight, respectively. Let Sn be the symmetric group of permutations on n elements. A binary code C of length n is said to be propelinear [8] if for any codeword x ∈ C there is a coordinate permutation πx ∈ Sn verifying the properties: x + πx (y) ∈ C if y ∈ C, πx ◦ πy = πz ∀y ∈ C, where z = x + πx (y). (1) * This work has been partially supported by Spanish CICYT grant TIC2003-08604-C04-01 and Catalan DURSI grant SGR2005-00319. 171 172 Borges, J. et al. If C is a propelinear code, we can define the binary operation ⋆ : C × F −→ F such that x ⋆ y = x + πx (y) ∀x ∈ C ∀y ∈ F . A binary code C with this operation, (C, ⋆), is a Hamming-compatible group code [8, 7, 2]. Propelinear codes include linear, Z4 -linear, Z2k -linear codes, etc. (see [8, 7, 2]), and they also include codes that had not been regarded as group codes, such as the original Preparata code [7]. A standard way to define a binary group code is by considering a group G such as Znk (the most studied case is when k = 4 [6]), Zn2k (see [4]) or, more generally, Zki11 × · · · × Zkirr . Then, we take a subgroup C ≤ G and, finally, the binary group code is the binary image Φ(C ), where Φ is a Gray map. That is, Φ is the componentwise extension of a one-to-one map φ : Z p ֒→ Zm 2 . If we only consider Gray maps with the classical property that d(φ (i), φ (i + 1)) = 1, then we can prove that if G is of the form Zki11 × · · · × Zkirr , then i1 , . . . , ir are all even [2]. Moreover, the binary code Φ(C ) is always a propelinear code. 2. Translation invariant propelinear codes As it is defined in [7], a propelinear code (C, ⋆) is said to be a translation invariant code if d(x, y) = d(x ⋆ u, y ⋆ u) ∀x, y ∈ C ∀u ∈ F . The class of translation invariant propelinear codes includes linear and Z 4 -linear codes. It is also proved in [7] that any translation invariant propelinear code of β length n can be viewed as a group isomorphic to a subgroup of Z α2 × Z4 × Q8κ , where α + 2β + 4κ = n and Q8 is the quaternion group on eight elements. Thereβ fore, translation invariant Abelian codes are subgroups of Zα2 × Z4 . In [2], it was also shown that if C ≤ G, where G = Zk2i11 × · · · × Zk2ir r and C = Φ(C ) is translation invariant or it is a perfect single error-correcting code, then β i j ≤ 2, for all j = 1, . . . , r. That is, C ≤ Zα2 × Z4 . 3. Z2 Z4 -linear codes β From now on, we will consider subgroups C of Zα2 × Z4 and the corresponding binary codes C = Φ(C ). We will take the following extension of the usual Gray β map: Φ : Zα2 × Z4 −→ Zn2 , where n = α + 2β , given by β Φ(x, y) = (x, φ (y)) ∀x ∈ Zα2 , ∀y ∈ Z4 ; On Z2 Z4 -linear codes and duality β 173 2β where φ : Z4 −→ Z2 is the usual Gray map, that is, φ (y1 , . . . , yβ ) = (ϕ (y1 ), . . . , ϕ (yβ )), and ϕ (0) = (0, 0), ϕ (1) = (0, 1), ϕ (2) = (1, 1), ϕ (3) = (1, 0). β Since C is a subgroup of Zα2 ×Z4 , it is also isomorphic to an Abelian structure γ like Z2 × Zδ4 . Therefore, we have that |C | = 2γ 4δ and the number of order two codewords in C is 2γ +δ . Note that the length of the binary code C = Φ(C ) is n = α + 2β . Let X (respectively Y ) be the set of Z2 (respectively Z4 ) coordinate positions, so |X| = α and |Y | = β . Call CX (respectively CY ) the code C restricted to the X (respectively Y ) coordinates. Let D be the subcode of C which contains all order two codewords and let k be the dimension of DX , which is a binary linear code. For the case α = 0, we will write k = 0. With all this in mind, we will say that C (or equivalently C = Φ(C )) is of type (α , β ; γ , δ ; k). β Definition 1. An additive code C is a subgroup of Zα2 × Z4 . We say that the binary image C = Φ(C ) is a Z2 Z4 -linear code of length n = α + 2β and type (α , β ; γ , δ ; k), where k is the dimension of DX and γ , δ defined as above. In [1], it was shown that this definition is equivalent to the one given in [3], when we consider the particular case of the Hamming metric association scheme. Note that the Z2 Z4 -linear codes are a generalization of binary linear codes and Z4 -linear codes. When β = 0, the binary code C = C corresponds to a binary linear code. On the other hand, when α = 0, the additive code C is a quaternary linear code and its corresponding binary code C = Φ(C ) is a Z4 -linear code. 4. Z2 Z4 -duality β We will use the following definition of the inner product in Zα2 × Z4 : α hu, vi = 2( ∑ ui vi ) + i=1 β α +β ∑ u j v j ∈ Z4 , (2) j=α +1 where u, v ∈ Zα2 × Z4 . Note that when α = 0 the inner product is the usual one for Z4 -vectors (i.e. vectors over Z4 ) and when β = 0 it is twice the usual one for Z2 -vectors. We can also write hu, vi = u·In ·vT , (3) 174 Borges, J. et al.  2Idα 0 is a diagonal quaternary matrix. where In = 0 Idβ Let C be a Z2 Z4 -linear code of length n of type (α , β ; γ , δ ; k). We denote by C the corresponding additive code, i.e., C = Φ−1 (C). The additive dual code of C , denoted by C ⊥ , is defined in the standard way  β C ⊥ = {u ∈ Zα2 × Z4 | hu, vi = 0 for all v ∈ C }. The corresponding binary code Φ(C ⊥ ) is denoted by C⊥ and called the Z2 Z4 -dual code of C. The additive dual code C ⊥ is also an additive code, that is a subgroup of β Zα2 × Z4 . Its weight enumerator polynomial is related to the weight enumerator polynomial of C by McWilliams Identity (see [5]). Notice that C and C⊥ are not dual in the binary linear sense but the weight enumerator polynomial of C⊥ is the McWilliams transform of the weight enumerator polynomial of C. Using this fact, we can prove easily that |C ||C ⊥ | = 2n . One could think on additive codes (or Z2 Z4 -linear codes) only as quaternary linear codes (or Z4 -linear codes), changing ones by twos in the coordinates over Z2 . However, it is not so simple. Take, for example, α = β = 1 and the vectors v = (1, 3) and w = (1, 2). It is easy to check that hv, wi = 0, so v and w are Z 2 Z4 orthogonal. If we change the ones by twos in the coordinates over Z 2 of these vectors we get v′ = (2, 3) and w′ = (2, 2), which are not orthogonal in the quaternary sense. β Since C ≤ Zα2 × Z4 , the code C could be seen as the kernel of a group homo′ γ′ morphism onto Z2 × Z4δ , that is, C = ker ϑ , where β γ′ ′ ϑ : Zα2 × Z4 −→ Z2 × Z4δ . The additive dual code C ⊥ is also the kernel of another group homomorphism γ onto Z2 × Zδ4 , that is, C ⊥ = ker ϑ ′ , where β γ ϑ ′ : Zα2 × Z4 −→ Z2 × Zδ4 . Although C is not a free module, the elements in the code C can be represenγ ted as ∑ λi ui + i=1 γ +δ ∑ µ j v j , where λi ∈ Z2 for 1 ≤ i ≤ γ , µ j ∈ Z4 for γ + 1 ≤ j ≤ j=γ +1 β γ + δ and ui , v j are vectors of order two and order four in Zα2 × Z4 , respectively. The vectors ui , v j give us a generator (γ + δ ) × (α + β )-matrix G for the code C . We can write G as   M1 2M2 G= , (4) M3 Q On Z2 Z4 -linear codes and duality 175 where M1 , M2 , M3 are matrices over Z2 of size γ × α , γ × β and δ × α , respectively; and Q is a matrix over Z4 of size δ × β with quaternary row vectors of order four. The homomorphism ϑ can be represented by a matrix G′ , which can be viewed as a parity check matrix for the additive code C or as a generator matrix for its additive dual code C ⊥ . Vice versa, the homomorphism ϑ ′ can be represented by a matrix G, which can be viewed as a parity check matrix for the additive dual code C ⊥ or as a generator matrix for the code C . In general, as we will see below, if α = 0, i.e. C and C⊥ are Z4 -linear codes, then γ = γ ′ (see also [6]). However, this is not true for Z2 Z4 -linear codes. Now, look at the following maps: ξ , from Z4 to Z2 , which is the usual one modulo two (ξ (0) = 0, ξ (1) = 1, ξ (2) = 0, ξ (3) = 1); χ , from Z 2 to Z4 , which is the usual inclusion from the additive structure in Z2 to Z4 (χ (0) = 0, χ (1) = 2); and ι , that is the identity from Z2 to Z4 (ι (0) = 0, ι (1) = 1). These maps can be β β extended to the maps (ξ , Id) : Zα4 × Z4 −→ Zα2 × Z4 and (χ , Id), (ι , Id) : Zα2 × β β Z4 −→ Zα4 × Z4 , which will also be denoted by ξ , χ and ι , respectively. Proposition 1. Let C be a Z2 Z4 -linear code of type (α , β ; γ , δ ; k) and C its corresponding additive code. Then, C ⊥ = ξ (χ (C )⊥ ). (5) The next lemma is a well-known result for Z4 -linear codes. Lemma 1. [6] If C is a Z4 -linear code of type (0, β ; γ , δ ; 0), then the Z4 -dual code C⊥ is of type (0, β ; γ , β − γ − δ ; 0). The next theorem relates the parameters of a Z2 Z4 -linear code and those of its Z2 Z4 -dual code. Theorem 1. Let C be a Z2 Z4 -linear code of type (α , β ; γ , δ ; k). The Z2 Z4 -dual code C⊥ is then of type (α , β ; γ ′ , δ ′ ; k′ ), where γ ′ = α + γ − 2k, δ ′ = β − γ − δ + k, k′ = α − k. Finally, we have obtained a standard form for the generator matrix of the additive dual code. Let ν ∈ Sn be the permutation ν = (1 n)(2 n − 1)(3 n − 2) · · · (⌊ n2 ⌋ n − ⌊ 2n ⌋ + 1) 176 Borges, J. et al. Proposition 2. Let C be an additive code of type (α , β ; γ , δ ; k). The code C is permutation-equivalent to an additive code with a generator matrix ξ (ν (G)), where G is a matrix of the form G=  0 Idδ 2Idγ R 2T S  (6) , where R, T are matrices over Z2 of size δ × γ and γ ×(α + β − δ − γ ) respectively, and S is a matrix over Z4 of size δ × (α + β − δ − γ ). The first α columns of ξ (ν (G)) correspond to the binary coordinates of C . Proposition 3. Let C be an additive code with generator matrix G as in (6). Then, ξ (ν (H)) is the generator matrix of C ⊥ , where H=  2Rtr 2Idγ
tr − S+TR T tr 0 Idα +β −γ −δ  , (7) where the first α columns of ξ (ν (H)) correspond to the binary coordinates of C ⊥. References [1] J. Borges and J. Rifà, “A characterization of 1-perfect additive codes”, IEEE Trans. Information Theory, vol. 45(5), pp. 1688-1697, 1999. [2] J. Borges, C. Fernandez and J. Rifà, “Every Z2k -code is a binary propelinear code”, In COMB’01. Electronic Notes in Discrete Mthematics, vol. 10, Elsevier Science, November 2001. [3] A.E. Brouwer, A.M. Cohen and A. Neumaier, Distance Regular Graphs. Springer-Verlag, 1989. [4] C. Carlet, “Z2k -Linear codes”, IEEE Trans. on Information Theory, vol. 44, pp. 1543-1547, 1998. [5] P. Delsarte, “An algebraic approach to the association schemes of coding theory”, Philips Research Rep. Suppl., vol. 10, 1973. [6] A.R. Hammons, P.V. Kumar, A.R. Calderbank, N.J.A. Sloane and P. Solé, “The Z4 -linearity of kerdock, preparata, goethals and related codes”, IEEE Trans. on Information Theory, vol. 40, pp. 301-319, 1994. On Z2 Z4 -linear codes and duality 177 [7] J. Pujol and J. Rifà, “Translation invariant propelinear codes”, IEEE Trans. Information Theory, vol. 43, pp. 590-598, 1997. [8] J. Rifà, J.M. Basart and L. Huguet, “On completely regular propelinear codes”, in Proc. 6th International Conference, AAECC-6. 1989, number 357 in LNCS, pp. 341-355, Springer-Verlag.