Left Dihedral Codes over Finite Chain Rings

TL;DR

This paper characterizes left ideals in dihedral group rings over finite chain rings, exploring their structure, duals, and self-dual codes, with specific classifications over Galois fields.

Contribution

It provides a complete characterization of left dihedral codes over finite chain rings and classifies such codes over Galois fields, including dual and self-dual codes.

Findings

Complete structure of left dihedral codes over finite chain rings.

Classification of codes over Galois fields for any positive integer N.

Identification of dual and self-dual codes within these classes.

Abstract

Let $R$ be a finite commutative chain ring, $D_{2n}$ be the dihedral group of size $2n$ and $R[D_{2n}]$ be the dihedral group ring. In this paper, we completely characterize left ideals of $R[D_{2n}]$ (called left $D_{2n}$-codes) when ${\rm gcd}(char(R),n)=1$. In this way, we explore the structure of some skew-cyclic codes of length 2 over $R$ and also over $R\times S$, where $S$ is an isomorphic copy of $R$. As a particular result, we give the structure of cyclic codes of length 2 over $R$. In the case where $R=\F_{p^m}$ is a Galois field, we give a classification for left $D_{2N}$-codes over $\F_{p^m}$, for any positive integer $N$. In both cases we determine dual codes and identify self-dual ones.