ON SOME CHARACTERIZATIONS OF GENERALIZED QUASI-CYCLIC CODES OVER Zq

Article Type

Research Article

Publication Title

Advances in Mathematics of Communications

Abstract

Quasi-cyclic (QC) codes are a natural and remarkable generalization of cyclic codes. QC codes are further generalized into what are known as generalized quasi-cyclic (GQC) codes. In this paper we study GQC codes of index 2 over Zq, where q = pm is a prime power, by considering them as a special case of mixed alphabet codes. Following some recent works on codes over mixed alphabets, GQC codes of index 2 over Zq can be seen as Zq-double cyclic codes. Such codes can be viewed as Zq[x]-submodules of Zq[x]/(xr − 1) × Zq[x]/(xs − 1). We determine explicitly the generator polynomials of this family of codes for two separate cases (r, q) = 1, (s, q) = 1 and (r, q) 6= 1, (s, q) 6= 1, and have given different characterizations of these codes. We have also obtained a minimal generating set for these codes. A new Gray map has been defined over Zq and some optimal p-ary linear and nonlinear codes have been obtained through this Gray map.

First Page

284

Last Page

303

DOI

10.3934/amc.2023059

Publication Date

2-1-2025

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