Linear complementary pair of group codes over finite principal ideal rings

Abstract

A pair $(C,D)$ of group codes over group algebra $R[G]$ is called a linear complementary pair (LCP) if $C\oplus D=R[G]$, where $R$ is a finite principal ideal ring, and $G$ is a finite group. We provide a necessary and sufficient condition for a pair $(C,D)$ of group codes over group algebra $R[G]$ to be LCP. Then we prove that if $C$ and $D$ are both group codes over $R[G]$, then $C$ and $D^{\perp }$ are permutation equivalent.