On LCP codes over a mixed ring alphabet

TL;DR

This paper studies linear codes over mixed finite chain rings, introducing a generator matrix, analyzing LCP properties, and establishing separability and permutation equivalence of certain code pairs.

Contribution

It provides a standard generator matrix for mixed-alphabet codes and proves new properties of LCP and product group codes over finite chain rings.

Findings

LCP codes over mixed rings are weakly-free.

Mixed-alphabet product group codes are separable.

LCP pairs imply permutation equivalence of codes and duals.

Abstract

In this paper, we introduce a standard generator matrix for mixed-alphabet linear codes over finite chain rings. Furthermore, we show that, when one has a linear complementary pair (LCP) of mixed-alphabet linear codes, both codes are weakly-free. Additionally, we establish that any mixed-alphabet product group code is separable. Thus, if one has a pair $\{C, D\}$ of mixed-alphabet product group codes over a finite chain ring that forms a LCP, it follows that $C$ and the Euclidean dual of $D$ are permutation equivalent.