On the $\sigma$ duals and $\sigma$ hulls of linear codes

Abstract

Let $\text{SLAut}(\mathbb{F}_{q}^{n})$ 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 some general properties of linear codes associated with the $\sigma$ duals for $\sigma \in \text{SLAut}(\mathbb {F}_{q}^{n})$. 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 the $\sigma$ hull of a linear code can be determined by a generator matrix of it or its $\sigma$ dual. We give a characterization on the $\sigma$ dual and $\sigma$ hull of a matrix-product code. We give an explanation for why it is meaningful to extend the $\ell$-Galois hulls of matrix-product codes to the $\sigma$ hulls of matrix-product codes. We also investigate the intersection of a pair of matrix-product codes. Finally, we construct several families of $\sigma$ dual-containing matrix-product codes, some of which are optimal or almost optimal according to the Database (Grassl 2023).