Linear and cyclic codes over direct product of finite chain rings

Abstract

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.

On $Z_{p^r}Z_{p^r}Z_{p^s}$-Additive Cyclic Codes

“…[9,4,4] (7, 1, 7) A(x) = x 7 + 1, B(x) = x + 1, l(x) = 0, l 1 (x) = x 4 + x 3 + x 2 + 1, l 2 (x) = 0, G(x) = x 4 + x 3 + x 2 + 1. [15,3,8] (1, 1, 15) A(x) = x + 1, B(x) = x + 1, l(x) = 0, l 1 (x) = 1, l 2 (x) = 1, G(x) = x 8 + x 7 + x 6 + x 4 + 1. [17,7,6] (1, 1, 15) A(x) = x + 1, B(x) = x + 1, l(x) = 0, l 1 (x) = 1, l 2 (x) = 1, G(x) = x 4 + x + 1.…”

“…After that, in 2016, Borges et al [13] studied the generating polynomials for dual codes of Z 2 Z 4 -additive cyclic codes. Very recently, Borges et al [15] studied binary images of additive cyclic codes over the product of chain rings. Some generalizations of Z 2 Z 4 -additive codes and other related codes can be seen in [5,6,8,9].…”

Galois LCD codes over mixed alphabets

“…Linear codes over a mixed alphabet over a finite chain ring have become a great research avenue in coding theory, see for example [1,2,3,4,5,7,8,12]. In [7], Borges et al were the pioneers in studying the algebraic structure of Z 2 Z 4 -additive codes as Z 4 -submodules (additive groups) of Z α 2 × Z β 4 , where α and β are two positive integers.…”

“…Later, Aydogdu and Siap generalized these additive codes to codes over Z 2 Z 2 s in [1] and over Z p r Z p s in [3] where r and s are positive integers, p is a prime number and 1 ≤ r ≤ s. Note that the last condition implies that the ring Z p r of integers modulo p r is the homomorphic image of the ring Z p s of integers modulo p s . A general approach for codes over a mixed alphabet over a finite chain ring is explored in [8] by J. Borges et al, they called them S 1 S 2 -linear codes, where S 1 and S 2 are finite chain rings such that S 1 is the homomorphic image of S 2 by a ring epimorphism.…”

Constacyclic Codes over Commutative Finite Principal Ideal Rings

“…Especially, cyclic and constacyclic codes over finite chain rings attracted a lot of attention, e.g., [11,14,25]. If the code length n is coprime to the characteristic of the finite chain ring R, [8] showed that the cyclic and negacyclic codes over R are principal; and a kind of generator polynomials for such codes are exhibited in [8,Theorem 3.6], and in [1,Theorem 2.4] (based on [8]) also. As for constacyclic codes, there are a lot of works varied from case to case: over particular finite chain rings R, for particular λ ∈ R and particular code length n; e.g., [2,3,6,7,9,20,21,22].…”