2018/09/21 by Hongwei Liu, Xu Pan, Liu, Hongwei +1 · 1 citation
Computer Science · #Coding theory and cryptography #Cooperative Communication and Network Coding #Error Correcting Code Techniques #FOS: Computer and information sciences #Information Theory (cs.IT)
paper · pdf · doi:10.48550/arxiv.1809.08053
openalex publication_date 2018/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The ℓ-Galois hull hℓ(C) of an [n,k] linear code C over a finite field \mathbbFq is the intersection of C and C^\botℓ, where C^\botℓ 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 hℓ(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. %The dimension of l-Galois hull of a code is not invariant under monomial equivalence if q>4. We show that every [n,k] linear code over \mathbb 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 \mathbb 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=pe for some prime p. As an application, we characterize the ℓ-Galois hull of matrix product codes over finite fields.