Galois Hulls of Linear Codes over Finite Fields

The $\ell$-Galois hull $h_{\ell}(C)$ of an $[n,k]$ linear code $C$ over a finite field $\mathbb{F}_q$ is the intersection of $C$ and $C^{{\bot}_{\ell}}$, where $C^{\bot_{\ell}}$ denotes the $\ell$-Galois dual of $C$ which introduced by Fan and Zhang (2017). The $\ell$- Galois LCD code is a linear code $C$ with $h_{\ell}(C) = 0$. In this paper, we show that the dimension of the $\ell$-Galois hull of a linear code is invariant under permutation equivalence and we provide a method to calculate the dimension of the $\ell$-Galois hull by the generator matrix of the code. Moreover, we obtain that the dimension of the $\ell$-Galois hulls of ternary codes are also invariant under monomial equivalence. %The dimension of $l$-Galois hull of a code is not invariant under monomial equivalence if $q>4$. We show that every $[n,k]$ linear code over $\mathbb F_{q}$ is monomial equivalent to an $\ell$-Galois LCD code for any $q>4$. We conclude that if there exists an $[n,k]$ linear code over $\mathbb F_{q}$ for any $q>4$, then there exists an $\ell$-Galois LCD code with the same parameters for any $0\le \ell\le e-1$, where $q=p^e$ for some prime $p$. As an application, we characterize the $\ell$-Galois hull of matrix product codes over finite fields.