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 $F_{p^e}$ of characteristic p. Let G be a generator polynomial matrix (GPM) of an MT code $C$. For any $0\le \kappa <e$, the κ-Galois hull of $C$, denoted by $h_{\kappa }\left(C\right)$, is the intersection of $C$ with its κ-Galois dual. The main result in this paper is that a GPM for $h_{\kappa }\left(C\right)$ has been obtained from G. We start by associating a linear code $Q_G$ with G. We show that $Q_G$ is quasi-cyclic. In addition, we prove that the dimension of $h_{\kappa }\left(C\right)$ is the difference between the dimension of $C$ and that of $Q_G$. Thus the determinantal divisors are used to derive a formula for the dimension of $h_{\kappa }\left(C\right)$. Finally, we deduce a GPM formula for $h_{\kappa }\left(C\right)$. In particular, we handle the cases of κ-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 codes with the best-known parameters, are used to illustrate the theoretical results.