Galois hulls of linear codes over finite fields

The l-Galois hull h(l)(C) of an [n, k] linear code C over the finite field F-q is the intersection of C and C-perpendicular to l, where C-perpendicular to l denotes the l-Galois dual of C which was introduced by Fan and Zhang in 2017. The l-Galois LCD code is a linear code C satisfying h(l)(C) = C boolean AND C-perpendicular to l = {0}. In this paper, we show that the dimension of the l-Galois hull of a linear code is invariant under permutation equivalences and we provide a method to calculate the dimension of the l-Galois hull through a generator matrix of the code. Moreover, we obtain that the dimension of the l-Galois hull of a ternary code is also invariant under monomial equivalences. We show that every [n, k] linear code over Fq is monomial equivalent to an l-Galois LCD code for any q > 4. We conclude that if there exists an [n, k] linear code over Fq for any q > 4, then there exists an l-Galois LCD code with the same parameters for any 0 <= l <= e - 1, where q = p(e) for some prime number p. As an application, we characterize the l-Galois hulls of matrix product codes over finite fields.