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

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.