Let \(q=p^{e}\) be a prime power and \(\ell \) be an integer with \(0 \le \ell < e\) . \(\ell \) -Galois self-orthogonal codes generalize both Euclidean self-orthogonal codes ( \(\ell =0\) ) and Hermitian self-orthogonal codes ( \(\ell =\frac{e}{2}\) and e is even). In this paper, we characterize two general methods for constructing \(\ell \) -Galois self-orthogonal matrix-product codes over \(\mathbb {F}_{q}\) from some non-singular matrices and special input codes, and determine the parameters of the obtained \(\ell \) -Galois self-orthogonal matrix-product codes when the underlying matrices and input codes satisfy some concrete conditions. Moreover, some special matrices used in the matrix-product constructions are also proposed. As an application, many quantum error-correcting codes (QECCs) with better parameters than the QECCs available in the literature are derived from the Hermitian self-orthogonal matrix-product codes.
