Generator polynomial matrices of the Galois hulls of multi-twisted codes

In this study, we consider the Euclidean and Galois hulls of multi-twisted (MT) codes over a finite field ${\mathbb{F}}_{p^e}$ of characteristic $p$. Let $\mathbf{G}$ be a generator polynomial matrix (GPM) of a MT code $\mathcal{C}$. For any $0\le \kappa <e$, the $\kappa$-Galois hull of $\mathcal{C}$, denoted by $h_{\kappa }\left(\mathcal{C}\right)$, is the intersection of $\mathcal{C}$ with its $\kappa$-Galois dual. The main result in this paper is that a GPM for $h_{\kappa }\left(\mathcal{C}\right)$ has been obtained from $\mathbf{G}$. We start by associating a linear code ${\mathcal{Q}}_{\mathbf{G}}$ with $\mathbf{G}$. We show that ${\mathcal{Q}}_{\mathbf{G}}$ is quasi-cyclic. In addition, we prove that the dimension of $h_{\kappa }\left(\mathcal{C}\right)$ is the difference between the dimension of $\mathcal{C}$ and that of ${\mathcal{Q}}_{\mathbf{G}}$. Thus the determinantal divisors are used to derive a formula for the dimension of $h_{\kappa }\left(\mathcal{C}\right)$. Finally, we deduce a GPM formula for $h_{\kappa }\left(\mathcal{C}\right)$. In particular, we handle the cases of $\kappa$-Galois self-orthogonal and linear complementary dual MT codes; we establish equivalent conditions that characterize these cases. Equivalent results can be deduced immediately for the classes of cyclic, constacyclic, quasi-cyclic, generalized quasi-cyclic, and quasi-twisted codes, because they are all special cases of MT codes. Some numerical examples, containing optimal and maximum distance separable codes, are used to illustrate the theoretical results.