The Hulls of Matrix-Product Codes over Commutative Rings and Applications

TL;DR

This paper studies the hulls of matrix-product codes over commutative rings, providing conditions for these codes to be LCD, and explores applications to torsion codes over finite chain rings with illustrative examples.

Contribution

It introduces new conditions for matrix-product codes over rings to be LCD and applies these to torsion codes over finite chain rings, expanding the understanding of code hulls.

Findings

Established sufficient conditions for matrix-product codes to be LCD.

Analyzed hull properties of codes over finite chain rings.

Provided examples illustrating the theoretical results.

Abstract

Given a commutative ring $R$ with identity, a matrix $A\in M_{s\times l}(R)$, and $R$-linear codes $\mathcal{C}_1, \dots, \mathcal{C}_s$ of the same length, this article considers the hull of the matrix-product codes $[\mathcal{C}_1 \dots \mathcal{C}_s]\,A$. Consequently, it introduces various sufficient conditions under which $[\mathcal{C}_1 \dots \mathcal{C}_s]\,A$ is a linear complementary dual (LCD) code. As an application, LCD matrix-product codes arising from torsion codes over finite chain rings are considered. Highlighting examples are also given.