Linear complementary pair of group codes over finite principal ideal rings

TL;DR

This paper characterizes linear complementary pairs of group codes over finite principal ideal rings and shows their permutation equivalence with dual codes.

Contribution

It provides a necessary and sufficient condition for a pair of group codes to be LCP over group algebras of finite principal ideal rings.

Findings

Characterization of LCP pairs over finite principal ideal rings.

Proof that group codes and their duals are permutation equivalent.

Establishment of conditions for code complementarity over group algebras.

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.