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 $C.$