Galois Hulls of Linear Codes over Finite Fields
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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. monomial equivalence if q>4. 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.
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