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.

翻译：本研究考虑特征为$p$的有限域$\mathbb{F}_{p^e}$上多扭转码的欧几里得壳与伽罗瓦壳。设$\mathbf{G}$为MT码$\mathcal{C}$的生成多项式矩阵（GPM）。对于任意$0\le \kappa<e$，$\mathcal{C}$的$\kappa$-伽罗瓦壳$h_\kappa\left(\mathcal{C}\right)$定义为$\mathcal{C}$与其$\kappa$-伽罗瓦对偶的交集。本文的主要结果是从$\mathbf{G}$出发得到了$h_\kappa\left(\mathcal{C}\right)$的GPM。我们首先将线性码$\mathcal{Q}_\mathbf{G}$与$\mathbf{G}$相关联，证明了$\mathcal{Q}_\mathbf{G}$为准循环码。进一步，我们证明$h_\kappa\left(\mathcal{C}\right)$的维数等于$\mathcal{C}$的维数与$\mathcal{Q}_\mathbf{G}$的维数之差。因此，利用行列式因子推导出$h_\kappa\left(\mathcal{C}\right)$维数的计算公式。最后，我们推导出$h_\kappa\left(\mathcal{C}\right)$的GPM公式。特别地，我们处理了$\kappa$-伽罗瓦自正交码和线性互补对偶MT码的情形，建立了刻画这些情形的等价条件。由于循环码、常循环码、准循环码、广义准循环码和准扭转码均为MT码的特例，因此可立即推导出这些码类的等价结论。文中通过包含最优码与最大距离可分码的数值算例对理论结果进行验证。