The ℓ-Galois hull h ℓ (C) of an [n, k] linear code C over a finite field F q is the intersection of C and C ⊥ ℓ , where C ⊥ ℓ denotes the ℓ-Galois dual of C which introduced by Fan and Zhang (2017). The ℓ-Galois LCD code is a linear code C with h ℓ (C) = 0. In this paper, we show that the dimension of the ℓ-Galois hull of a linear code is invariant under permutation equivalence and we provide a method to calculate the dimension of the ℓ-Galois hull by the generator matrix of the code. Moreover, we obtain that the dimension of the ℓ-Galois hulls of ternary codes are also invariant under monomial equivalence. We show that every [n, k] linear code over F q 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 F q 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 p. As an application, we characterize the ℓ-Galois hull of matrix product codes over finite fields.

In this paper, we introduce the notion of generalized pair weights of an [n, k]linear code over a finite field and the notion of pair r-equiweight codes, where 1 ≤ r ≤ k − 1. We give some properties of generalized pair weights of linear codes over finite fields. We obtain a necessary and sufficient condition for an [n, k]-linear code to be a pair equiweight code and characterize the pair r-equiweight codes for any 1 ≤ r ≤ k − 1. In addition, a relationship between the pair equiweight code and the pair r-equiweight code is also given. Finally, we prove the MacWilliams extension theorem for the pair weight case.

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