We propose a decoding algorithm for the (u | u + v)-construction that decodes up to half of the minimum distance of the linear code. We extend this algorithm for a class of matrix-product codes in two different ways. In some cases, one can decode beyond the error-correction capability of the code.
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good q-ary quantum codes of large length. Specifically, it follows from CSS construction (resp. Hermitian construction) that a matrix-product code over Fq (resp. Fq2) which is Euclidean dual-containing (resp. Hermitian dual-containing) can produce a q-ary quantum code. In order to obtain such matrix-product codes, a common way is to construct quasi-orthogonal matrices (resp. quasi-unitary matrices) as the defining matrices of matrix-product codes over Fq (resp. Fq2). The usage of NSC quasi-orthogonal matrices or NSC quasi-unitary matrices in this process enables the minimum distance lower bound of the corresponding quantum codes to reach its optimum. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in \cite{Cao2020Constructioncaowang,Cao2020New,Cao2020Constructionof}; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and q-ary quantum codes derived from these newly constructed matrix-product codes.