Linear complementary pairs of codes over a finite non-commutative Frobenius ring

TL;DR

This paper investigates the structure and properties of linear complementary pairs of codes over finite non-commutative Frobenius rings, providing conditions for their duals and exploring their security parameters.

Contribution

It offers a necessary and sufficient condition for LCPs over non-commutative Frobenius rings and proves dual code equivalence under certain conditions.

Findings

Characterization of LCPs over non-commutative Frobenius rings

Conditions for dual code equivalence in LCPs

Relation between minimum distances and security parameters

Abstract

In this paper, we study linear complementary pairs (LCP) of codes over finite non-commutative local rings. We further provide a necessary and sufficient condition for a pair of codes $(C,D)$ to be LCP of codes over finite non-commutative Frobenius rings. The minimum distances $d(C)$ and $d(D^\perp)$ are defined as the security parameter for an LCP of codes $(C, D).$ It was recently demonstrated that if $C$ and $D$ are both $2$-sided LCP of group codes over a finite commutative Frobenius rings, $D^\perp$ and $C$ are permutation equivalent in \cite{LL23}. As a result, the security parameter for a $2$-sided group LCP $(C, D)$ of codes is simply $d(C)$. Towards this, we deliver an elementary proof of the fact that for a linear complementary pair of codes $(C,D)$, where $C$ and $D$ are linear codes over finite non-commutative Frobenius rings, under certain conditions, the dual code $D^\perp$ is equivalent to $C.$