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.
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.
A Z 2 Z 4 -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 Z 2 coordinates and the set of Z 4 coordinates, such that any cyclic shift of the coordinates of both subsets leaves the code invariant. Let Φ(C) be the binary Gray map image of C. We study the rank and the dimension of the kernel of a Z 2 Z 4 -additive cyclic code C, that is, the dimensions of the binary linear codes Φ(C) and ker(Φ(C)). We give upper and lower bounds for these parameters. It is known that the codes Φ(C) and ker(Φ(C)) are binary images of Z 2 Z 4 -additive codes R(C) and K(C), respectively. Moreover, we show that R(C) and K(C) are also cyclic and we determine the generator polynomials of these codes in terms of the generator polynomials of the code C.Keywords Z 2 Z 4 -additive cyclic codes, Gray map, kernel, rank.
I. INTRODUCTIONDenote 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
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