Let $\mathrm{SLAut}(\mathbb{F}_{q}^{n})$ denote the group of all semilinear isometries on $\mathbb{F}_{q}^{n}$, where $q=p^{e}$ is a prime power. In this paper, we investigate general properties of linear codes associated with $\sigma$ duals for $\sigma\in\mathrm{SLAut}(\mathbb{F}_{q}^{n})$. We show that the dimension of the intersection of two linear codes can be determined by generator matrices of such codes and their $\sigma$ duals. We also show that the dimension of $\sigma$ hull of a linear code can be determined by a generator matrix of it or its $\sigma$ dual. We give a characterization on $\sigma$ dual and $\sigma$ hull of a matrix-product code. We also investigate the intersection of a pair of matrix-product codes. We provide a necessary and sufficient condition under which any codeword of a generalized Reed-Solomon (GRS) code or an extended GRS code is contained in its $\sigma$ dual. As an application, we construct eleven families of $q$-ary MDS codes with new $\ell$-Galois hulls satisfying $2(e-\ell)\mid e$, which are not covered by the latest papers by Cao (IEEE Trans. Inf. Theory 67(12), 7964-7984, 2021) and by Fang et al. (Cryptogr. Commun. 14(1), 145-159, 2022) when $\ell\neq \frac{e}{2}$.
翻译:让$\mathrm{SLAut}(\mathbb{F}_{q}^{n})$表示所有$\mathbb{F}_{q}^{n}$上的半线性同构的群,其中$q=p^{e}$为素数幂。在本文中,我们研究与$\sigma\in\mathrm{SLAut}(\mathbb{F}_{q}^{n})$的$\sigma$对偶相关的线性码的一般性质。我们表明,两个线性码的交集的维数可以通过这样的码及其$\sigma$对偶的生成矩阵来确定。我们还表明,线性码的$\sigma$壳的维数可以通过其生成矩阵或其$\sigma$对偶来确定。我们给出了矩阵积码的$\sigma$对偶和$\sigma$壳的描述。我们还研究了一对矩阵积码的交集。我们提供了广义Reed-Solomon(GRS)码或扩展GRS码的任意码字所包含的$\sigma$对偶的必要和充分条件。作为一种应用,当$2(e-\ell)\mid e$时,我们构造了11个新的$q$进制MDS码家族,这些码具有满足Cao(IEEE Trans. Inf. Theory 67(12),7964-7984,2021)和Fang等人(Cryptogr. Commun. 14(1), 145-159, 2022)的最新论文未涵盖的$\ell$-Galois壳,其中$\ell\neq \frac{e}{2}$。