Duality on group algebras over finite chain rings: applications to additive group codes

Abstract

Given a finite group $G$ and an extension of finite chain rings $S|R$, one can consider the group rings $\mathscr{S} = S[G]$ and $\mathscr{R} = R[G]$. The group ring $\mathscr{S}$ can be viewed as an $R$-bimodule, and any of its $R$-submodules naturally inherits an $R$-bimodule structure; in the framework of coding theory, these are called \emph{additive group codes}, more precisely a (left) additive group code of is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism which maps $G$ to the standard basis of $S^n$, where $n=|G|$. In the first part of the paper, the ring extension $S|R$ is studied, and several $R$-module isomorphisms are established for decomposing group rings, thereby providing a characterization of the structure of additive group codes. In the second part, we construct a symmetric, nondegenerate trace-Euclidean inner product on $\mathscr{S}$. Two additive group codes $\mathcal{C}$ and $\mathcal{D}$ form an \emph{additive complementary pair} (ACP) if $\mathcal{C} + \mathcal{D} = \mathscr{S}$ and $\mathcal{C} \cap \mathcal{D} = \{0\}$. For two-sided ACPs, we prove that the orthogonal complement of one code under the trace-Euclidean duality is precisely the image of the other under an involutive anti-automorphism of $\mathscr{S}$, linking coding-theoretical ACPs with module orthogonal direct-sum decompositions, representation theory, and the structure of group algebras over finite chain rings.