The hulls of matrix-product codes over commutative rings and applications

Abstract

Given a commutative ring R with identity, a matrix $$A\in M_{s\times l}(R)$$ , and linear codes $$\mathcal {C}_1, \dots , \mathcal {C}_s$$ over R of the same length, this article considers the hull of the matrix-product code $$[\mathcal {C}_1 \dots \mathcal {C}_s]\,A$$ . Consequently, it introduces various sufficient conditions (as well as some necessary conditions in certain cases) under which $$[\mathcal {C}_1 \dots \mathcal {C}_s]\,A$$ is a complementary dual (LCD) code. As an application, LCD matrix-product codes arising from torsion codes over finite chain rings are considered. Moreover, we show the existence of asymptotically good sequences of LCD matrix-product codes over such rings.