  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
$\mathbb{F}_{q}$ (resp. $\mathbb{F}_{q^{2}}$) 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 $\mathbb{F}_{q}$ (resp.
$\mathbb{F}_{q^{2}}$). 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.
