The \(\ell \)-Galois hull \(h_{\ell }(C)\) of an [n, k] linear code C over the 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 was introduced by Fan and Zhang in 2017. The \(\ell \)-Galois LCD code is a linear code C satisfying \(h_{\ell }(C)= C\bigcap C^{\bot _{\ell }}= \{0\}\). In this paper, we show that the dimension of the \(\ell \)-Galois hull of a linear code is invariant under permutation equivalences and we provide a method to calculate the dimension of the \(\ell \)-Galois ...
