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^) 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^ and C are permutation equivalent in 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^ is equivalent to C.