A Chinese remainder theorem approach to skew generalized quasi-cyclic codes over finite fields

Abstract

In this work, we study a class of generalized quasi-cyclic (GQC) codes called skew GQC codes. By the factorization theory of ideals, we give the Chinese Remainder Theorem over the skew polynomial ring, which leads to a canonical decomposition of skew GQC codes. We also focus on some characteristics of skew GQC codes in details. For a 1-generator skew GQC code, we define the parity-check polynomial, determine the dimension and give a lower bound on the minimum Hamming distance. The skew quasi-cyclic (QC) codes are also discussed briefly.

A circulant approach to skew-constacyclic codes

“…This generalizes [5,Lem. 8] (see also [11,Thm. 2.1(iii)]), where a central polynomial x n − 1 is considered. In that case θ n is the identity on R and thus θ −n (h) = h. In particular, all of this generalizes the classical commutative case where h is the check polynomial of C [18, Ch.…”

“…Cyclic block codes form the most powerful class of linear block codes due to their inherent algebraic structure which allows the design of codes with large distance and efficient decoding algorithms. In recent years the notion of cyclicity has been generalized to skew-cyclicity, mainly in the work by Boucher/Ulmer and coworkers, see [3,5,10,6,7], but also by Abualrub et al [1], Matsuoka [19], and Gao et al [11].…”

Skew cyclic codes over $F_{q}+uF_{q}+vF_{q}+uvF_{q}$

“…In [1], T. Abualrub and P. Seneviratne studied skew cyclic codes over ring F 2 + vF 2 with v 2 = v. Moreover, J. Gao [6] and F. Gursoy et al [8] presented skew cyclic codes over F p + vF p and F q + vF q with different automorphisms, respectively. In [7], J. Gao et al also studied skew generalized quasi-cyclic codes over finite fields.…”

On the Codes over a Semilocal Finite Ring

“…Some authors generalized the notion of cyclic, quasi-cyclic and constacyclic codes by using generator polynomials in skew polynomial rings [1,2,5,7,8,9,14,15,18,27,30].…”