Left Dihedral Codes over Finite Chain Rings

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 $\mathrm{g}\mathrm{c}\mathrm{d}(char(R),n)=1$. 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 $R={\text{F}}_{p^m}$ is a Galois field, we give a classification for left $D_{2N}$-codes over ${\text{F}}_{p^m}$, for any positive integer $N$. In both cases we determine dual codes and identify self-dual ones.