${\mathbb {Z}}_{2}{\mathbb {Z}}_{4}$ -Additive Cyclic Codes, Generator Polynomials, and Dual Codes

Abstract

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.

Some results on <inline-formula><tex-math id="M1111">\begin{document}$ \mathbb{Z}_p\mathbb{Z}_p[v] $\end{document}</tex-math></inline-formula>-additive cyclic codes

“…, 4) [12,7,4] 3 [4,4] f Table 2. [3,3] f = x − 1, l = 1, g1 = 1, g2 = 1 (6, 3 8 , 2) 5 [4,4] f = x − 1, l = 1, g1 = 1, g2 = 1 (8, 5 11 , 2) 11 [7,8] f = x − 1, l = 1, g1 = 1, g2 = 1 (15, 11 22 , 2) 29 [12,6]…”

“…The most typical study was that the generator matrices and duality of Z 2 Z 4 -additive codes were determined [7]. Further, Abualrub et al and Borges et al studied Z 2 Z 4 -additive cyclic codes and their dual codes, respectively [1,8]. At almost the same time, encoders have also studied many additive codes over other rings [2,4,5].…”

Linear and cyclic codes over direct product of finite chain rings

“…We denote the polynomial by θ m ( x ). Using this notation, in Borges et al, it is proven that if , then x n m −1=( x n −1) θ m ( x n ). From now on, denotes the least common multiple of α and β .…”

“…These codes can be identified as submodules of the ‐module . The duality of these codes is studied in Borges et al Cyclic codes over the direct product of 2 finite rings are defined in the same way; that is, the coordinates are partitioned into 2 sets corresponding to the coordinates over the first ring and the coordinates over the second ring and the simultaneous shift of the coordinates in both sets of a codeword is also a codeword. Some examples of such codes are cyclic codes in given in Abualrub et al, cyclic codes in in Borges et al, cyclic codes in in Gao et al, and cyclic codes in in Yao et al Note that for a ring R , cyclic codes over the product ring R α × R β are called double cyclic codes over R and are not the same that cyclic codes over R α + β .…”

Multifractal characteristic-based comparison of elements in soils within the Daxing and Yicheng areas of Hefei, Anhui Province, China

“…A class of codes which contains all binary and quaternary codes as a subclass is called Z 2 Z 4 -additive codes. This class of codes have been studied in [4,5,7]. Z 2 Z 2 [u]-additive codes have been studied in [2].…”