A code C is Z 2 Z 4 -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y ) is a binary linear code (respectively, a quaternary linear code). In this paper Z 2 Z 4 -additive codes are studied. Their corresponding binary images, via the Gray map, are Z 2 Z 4 -linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are found and standard forms for generator and parity-check matrices are given. In order to do this, the appropriate concept of duality for Z 2 Z 4 -additive codes is defined and the parameters of their dual codes are computed.

A code C is Z 2 Z 4 -additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y ) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of Z 2 Z 4 -additive codes under an extended Gray map are called Z 2 Z 4 -linear codes. In this paper, the invariants for Z 2 Z 4 -linear codes, the rank and dimension of the kernel, are studied. Specifically, given the algebraic parameters of Z 2 Z 4 -linear codes, the possible values of these two invariants, giving lower and upper bounds, are established. For each possible rank r between these bounds, the construction of a Z 2 Z 4 -linear code with rank r is given. Equivalently, for each possible dimension of the kernel k, the construction of a Z 2 Z 4 -linear code with dimension of the kernel k is given. Finally, the bounds on the rank, once the kernel dimension is fixed, are established and the construction of a Z 2 Z 4 -linear code for each possible pair (r, k) is given.

A Z2Z4-additive code C ⊆ Z α 2 × Z β 4 is called cyclic if the set of coordinates can be partitioned into two subsets, the set of Z2 and the set of Z4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. These codes can be identified as submodules of the Z4[x]-module Z2[x]/(x α − 1) × Z4[x]/(x β − 1). The parameters of a Z2Z4-additive cyclic code are stated in terms of the degrees of the generator polynomials of the code. The generator polynomials of the dual code of a Z2Z4-additive cyclic code are determined in terms of the generator polynomials of the code C. Index Terms Binary cyclic codes, Cyclic codes over Z4, Duality, Z2Z4-additive cyclic codes. I. INTRODUCTION Denote by Z 2 and Z 4 the rings of integers modulo 2 and modulo 4, respectively. We denote the space of n-tuples over these rings as Z n 2 and Z n 4. A binary code is any non-empty subset C of Z n 2. If that subset is a vector space then we say that it is a linear code. A code over Z 4 is a non-empty subset C of Z n 4 and a submodule of Z n 4 is called a linear code over Z 4. In Delsarte's 1973 paper (see [5]), he defined 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 2 n , the only structures for the abelian group are those of the form Z α 2 × Z β 4 , with α + 2β = n. This means that the subgroups C of Z α 2 × Z β 4 are the only additive codes in a binary Hamming scheme. In [4], Z 2 Z 4-additive codes were studied. Manuscript received Month day, year; revised Month day, year.

Abstract$${\mathbb {Z}}_{p^s}$$ Z p s -additive codes of length n are subgroups of $${\mathbb {Z}}_{p^s}^n$$ Z p s n , and can be seen as a generalization of linear codes over $${\mathbb {Z}}_2$$ Z 2 , $${\mathbb {Z}}_4$$ Z 4 , or $${\mathbb {Z}}_{2^s}$$ Z 2 s in general. A $${\mathbb {Z}}_{p^s}$$ Z p s -linear generalized Hadamard (GH) code is a GH code over $${\mathbb {Z}}_p$$ Z p which is the image of a $${\mathbb {Z}}_{p^s}$$ Z p s -additive code by a generalized Gray map. In this paper, we generalize some known results for $${\mathbb {Z}}_{p^s}$$ Z p s -linear GH codes with $$p=2$$ p = 2 to any odd prime p. First, we show some results related to the generalized Carlet’s Gray map. Then, by using an iterative construction of $${\mathbb {Z}}_{p^s}$$ Z p s -additive GH codes of type $$(n;t_1,\ldots , t_s)$$ ( n ; t 1 , … , t s ) , we show for which types the corresponding $${\mathbb {Z}}_{p^s}$$ Z p s -linear GH codes of length $$p^t$$ p t are nonlinear over $${\mathbb {Z}}_p$$ Z p . For these codes, we compute the kernel and its dimension, which allow us to give a partial classification. The obtained results for $$p\ge 3$$ p ≥ 3 are different from the case with $$p=2$$ p = 2 . Finally, the exact number of non-equivalent such codes is given for an infinite number of values of s, t, and any $$p\ge 2$$ p ≥ 2 ; by using also the rank as an invariant in some specific cases.

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The Z2s -additive codes are subgroups of Z n 2 s , and can be seen as a generalization of linear codes over Z2 and Z4. A Z2s -linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s -additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the Z4-linear Hadamard codes. However, when s > 2, the dimension of the kernel of Z2slinear Hadamard codes of length 2 t only provides a complete classification for some values of t and s. In this paper, the rank of these codes is computed for s = 3. Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once t ≥ 3 is fixed. In this case, the number of nonequivalent such codes is also established.

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We introduce a new type of linear and cyclic codes. These codes are defined over a direct product of two finite chain rings. The definition of these codes as certain submodules of the direct product of copies of these rings is given and the cyclic property is defined. Cyclic codes can be seen as submodules of the direct product of polynomial rings. Generator matrices for linear codes and generator polynomials for cyclic codes are determined.