Reversible complement cyclic codes over finite chain rings

TL;DR

This paper investigates reversible complement cyclic codes over finite chain rings, providing necessary and sufficient conditions, and applies these to classify certain cyclic codes over a specific ring of length 4.

Contribution

It establishes necessary and sufficient conditions for (u,k) reversible complement cyclic codes over finite chain rings and classifies specific codes over Z_{2}+vZ_{2}+v^{2}Z_{2}.

Findings

Derived necessary and sufficient conditions for (u,k) reversible complement codes.

Classified all cyclic codes of length 4 over Z_{2}+vZ_{2}+v^{2}Z_{2} as reversible complement or not.

Identified specific codes that satisfy the reversible complement property.

Abstract

Let k be an arbitrary element of a finite commutative chain ring R and u be a unit in R. In this work, we present necessary conditions which are sufficient as well for a cyclic code to be a (u,k) reversible complement code over R. Using these conditions, all principally generated cyclic codes over the ring Z_{2}+vZ_{2}+v^{2}Z_{2}, v^{3}=0 of length 4 have been checked to find whether they are (1,1) reversible complement or not.