  In recent literature, matrix product (MP) codes and their duals have gained
significant attention due to their application in the construction of quantum
stabilizer codes. In this paper, we begin with providing a formula that
characterizes the Galois dual of MP codes. Using this formula, we establish the
conditions under which MP codes are self-orthogonal and dual-containing.
Although similar results may exist in the literature, the novelty and
superiority of our results can be identified in the following points. Previous
results that characterize the duals of MP codes only apply to MP codes with an
invertible square defining matrix $\mathcal{A}$. However, our characterization
applies to MP code with any defining matrix, whether $\mathcal{A}$ is not
square or not of full row rank. Previous studies on the conditions for
self-orthogonality or dual-containment of MP codes have assumed certain
structures for the product $\mathcal{A}\mathcal{A}^T$ or
$\mathcal{A}\mathcal{A}^{\dagger}$, such as being diagonal, anti-diagonal,
monomial, or partitioned Hermitian orthogonal. However, our conditions do not
necessitate such specific structures. Previous studies investigated MP code
duality in the context of Euclidean and Hermitian duals; however, we
investigate MP code duality in the broader context of Galois dual, with
Euclidean and Hermitian duals acting as special cases. Finally, it is worth
noting that the proposed conditions for Galois self-orthogonality or
dual-containment are both necessary and sufficient. To demonstrate the
theoretical results, several numerical examples with best-known parameters MP
codes are provided.
