Linear complementary pairs of codes over a finite non-commutative Frobenius ring
Sanjit Bhowmick, Xiusheng Liu
TL;DR
This work addresses the problem of linear complementary pairs ($C,D$) of codes over finite non-commutative Frobenius rings and the relation between a component and the dual of the other. It develops an algebraic framework using local Frobenius structure, the residue map to $\mathbb{F}_q$, and injective hulls to characterize LCPs, including when LCP is preserved under reduction modulo $J(R)$ and when inverse relationships hold via matrix invertibility conditions. Key contributions include a necessary and sufficient condition for LCP over finite non-commutative Frobenius rings, a demonstration that free (hence projective) codes arise in LCP settings, and explicit idempotent-based constructions proving that $C^\bot$ is equivalent to $D$ (and $D^\bot$ to $C$) under suitable hypotheses; the results extend known commutative Frobenius and group-code findings to the non-commutative case and provide non-free LCP examples with potential cryptographic relevance.
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 $C.$