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arXiv · 2406.15794

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

Sanjit Bhowmick·Xiusheng Liu
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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 $C.$

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Sanjit Bhowmick, Xiusheng Liu. 2024-06-22. Linear complementary pairs of codes over a finite non-commutative Frobenius ring. https://arxiv.org/abs/2406.15794

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