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.

给定具有标识的交换环R，矩阵\（A_in M_ {s \ times l}（R）\）和线性代码\（\ mathcal {C} _1，\ dots，\ mathcal {C} _s \ ）在相同长度的R上，本文考虑矩阵乘积代码\（[\ mathcal {C} _1 \ dots \ mathcal {C} _s] \，A \）的外壳。因此，它引入了各种充分条件（在某些情况下还包括某些必要条件），其中\（[\ mathcal {C} _1 \ dots \ mathcal {C} _s] \，A \）是互补的双重（LCD）代码。作为一种应用，考虑了由有限链环上的扭转代码产生的LCD矩阵产品代码。此外，我们表明在此类环上存在LCD矩阵乘积代码的渐近良好序列。