Galois hulls of linear codes over finite fields

Abstract

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.

A NOTE ON k-GALOIS LCD CODES OVER THE RING

“…Suppose e is even and k = e/2. Then for a linear code C of type (n; k 1 , k 2 , d) over R with p k > 2, there exists a Hermitian LCD code C ′ which is equivalent to C over R.In other words, Theorems 4.3 and 4.6 generalise the results of [1, Corollaries 13, 18], concerning the constructions of Euclidean and Hermitian LCD codes over F q , to the k-Galois LCD codes over the chain ring R. Furthermore, Theorems 4.3 and 4.6 also generalise the results of[7, Theorem 4.8] introduced in[1].…”

“…LetN = G(G (p e−k ) ) T andN ′ = G a (G (p e−k ) a ) T . ThenN ′ = G a (G (p e−k ) a ) T =N + diag k 1 +k 2 [u ′ ],160 R. Wu and M. Shi[7] where u ′ = (a p e−k +1 . .…”

Self-reciprocal and self-conjugate-reciprocal irreducible factors of x − λ and their applications

“…Recently, many scholars investigated Galois constacyclic codes over finite fields, see [3,15,25,26,27,29,30]. Motivated by their work, we introduce the definition of σself-reciprocal polynomials over finite fields.…”

Galois hulls of cyclic serial codes over a finite chain ring

“…The following proposition is a natural generalization to finite chain rings of [11,Proposition 2.2].…”