Matrix-Product Codes over Commutative Rings and Constructions Arising from $(\sigma,\delta)$-Codes

TL;DR

This paper extends the understanding of matrix-product codes over commutative rings, establishing bounds on their Hamming distance, conditions for freeness and duality, and introducing new constructions from $(\sigma,\delta)$-codes.

Contribution

It generalizes known bounds to all commutative rings, provides conditions for dual and free MPCs, and introduces novel MPC constructions from $(\sigma,\delta)$-codes.

Findings

Hamming distance bound holds over any commutative ring.

Conditions for MPC duals to be MPCs are established.

New MPCs constructed from $(\sigma,\delta)$-codes.

Abstract

A well-known lower bound (over finite fields and some special finite commutative rings) on the Hamming distance of a matrix-product code (MPC) is shown to remain valid over any commutative ring $R$. A sufficient condition is given, as well, for such a bound to be sharp. It is also shown that an MPC is free when its input codes are all free, in which case a generating matrix is given. If $R$ is finite, a sufficient condition is provided for the dual of an MPC to be an MPC, a generating matrix for such a dual is given, and characterizations of LCD, self-dual, and self-orthogonal MPCs are presented. Finally, results of this paper are used along with previous results of the authors to construct novel MPCs arising from $(\sigma, \delta)$-codes. Some properties of such constructions are also studied.