Linear complementary pairs of codes over a finite non-commutative Frobenius ring
<|reference_start|>Linear complementary pairs of codes over a finite non-commutative Frobenius ring: 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 $C.$<|reference_end|>
@article{bhowmick2024linear,
    title={Linear complementary pairs of codes over a finite non-commutative
  Frobenius ring},
    author={Sanjit Bhowmick, Xiusheng Liu},
    journal={arXiv preprint arXiv:2406.15794},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.15794},
    primaryClass={cs.IT math.IT}
}