Left dihedral codes over finite chain rings

Abstract

Let R be a finite commutative chain ring, D2n be the dihedral group of size 2n and R[D2n] be the dihedral group ring. In this paper, we completely characterize left ideals of R[D2n] (called left D2n-codes) when 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×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=Fpm is a Galois field, we give a classification for left D2N-codes over Fpm, for any positive integer N. In both cases we determine dual codes and identify self-dual ones.