Reversible cyclic codes over finite chain rings

TL;DR

This paper establishes conditions for reversibility of cyclic codes over finite chain rings, characterizes MDS reversible codes, and explores properties of torsion codes and code parameters.

Contribution

It provides necessary and sufficient conditions for reversibility, characterizes MDS reversible cyclic codes, and analyzes torsion codes over finite chain rings.

Findings

Reversibility conditions for cyclic codes over chain rings

Characterization of MDS reversible cyclic codes of length p^s

Torsion codes of reversible cyclic codes are reversible

Abstract

In this paper, necessary and sufficient conditions for the reversibility of a cyclic code of arbitrary length over a finite commutative chain ring have been derived. MDS reversible cyclic codes having length p^s over a finite chain ring with nilpotency index 2 have been characterized and a few examples of MDS reversible cyclic codes have been presented. Further, it is shown that the torsion codes of a reversible cyclic code over a finite chain ring are reversible. Also, an example of a non-reversible cyclic code for which all its torsion codes are reversible has been presented to show that the converse of this statement is not true. The cardinality and Hamming distance of a cyclic code over a finite commutative chain ring have also been determined.