Abstract
Besides being coding-theoretically very useful in their own right, Euclidean self-orthogonal and self-dual codes have proved to be interesting and usable in diverse areas of mathematics and its applications such as group theory, combinatorial designs, communication systems, and lattice theory (see [5, 6, 19, 20]). On the other hand, Blackmore and Norton, in their pioneering paper [2], introduced the important notion of matrix-product codes over finite fields. A matrix-product code utilizes a finite list of (input) codes of the same length to produce a longer code. The parameters and decoding capabilities of some of such codes were studied by many authors (see for instance [2, 9, 10]). Some authors also considered matrix-product codes and some of their properties over certain finite commutative rings (see for instance [1, 3, 4, 7]). To connect the aforementioned concepts, one proper question on the topic is, "when can one construct
Self-orthogonal codes and self-dual codes, on the one hand, and matrix-product codes, on the other, form important and sought-after classes of linear codes. Combining the two constructions would be advantageous. Adding to this combination the relaxation of the underlying algebraic structures to be commutative rings instead of fields would be even more advantageous. The current article paves a path in this direction. The authors study the problem of self-orthogonality and self-duality of matrix-product codes over a commutative ring with identity. Some methods as well as special matrices are introduced for the construction of such codes. A characterization of such codes in some cases is also given. Some concrete examples as well as applications to torsion codes are presented.