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

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 $C.$