Structure And Application Of L-galois Hulls Of The Linear Code

Let F.be a finite field with q elements and C?l be the l-Galois dual of a linear code C which introduced by Fan and Zhang(2017).The most crucial definition of this article is the l-Galois hull hl(C)of a linear code C which is the intersection of C and C?l.The l-Galois LCD code is a linear code C with he(C)? 0 and widely used in classic coding theory and quantum coding theory.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 by a generator matrix of the linear code.Moreover,we obtain that the dimension of the l-Galois hulls of a ternary codes is also invariant under monomial equivalences.We show that every[n,k]linear code over Fq is mono-mial 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 = pl for some prime p.As an application,we characterize the l-Galois hull of a matrix product codes over a finite fields.