Intersections of linear codes and related MDS codes with new Galois hulls

TL;DR

This paper explores properties of linear codes related to Galois hulls, providing new characterizations and constructions of MDS codes with specific Galois hull dimensions, expanding the understanding of code duality and intersections.

Contribution

It introduces new theoretical results on $\sigma$ duals and hulls of linear codes, and constructs eleven families of MDS codes with novel Galois hull properties not covered by prior work.

Findings

Dimension of code intersections can be determined by generator matrices and $\sigma$ duals.

Characterization of $\sigma$ hulls via generator matrices and $\sigma$ duals.

Construction of eleven new families of $q$-ary MDS codes with specific Galois hulls.

Abstract

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 $\ell\neq \frac{e}{2}$.