Intersections of linear codes and related MDS codes with new Galois hulls

10/11/2022

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by   Meng Cao, et al.

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Let SLAut(𝔽_q^n) denote the group of all semilinear isometries on 𝔽_q^n, where q=p^e is a prime power. In this paper, we investigate general properties of linear codes associated with σ duals for σ∈SLAut(𝔽_q^n). We show that the dimension of the intersection of two linear codes can be determined by generator matrices of such codes and their σ duals. We also show that the dimension of σ hull of a linear code can be determined by a generator matrix of it or its σ dual. We give a characterization on σ dual and σ hull of a matrix-product code. We also investigate the intersection of a pair of matrix-product codes. We provide a necessary and sufficient condition under which any codeword of a generalized Reed-Solomon (GRS) code or an extended GRS code is contained in its σ dual. As an application, we construct eleven families of q-ary MDS codes with new ℓ-Galois hulls satisfying 2(e-ℓ)| e, which cannot be produced by the latest papers by Cao (IEEE Trans. Inf. Theory 67(12), 7964-7984, 2021) and by Fang et al. (Cryptogr. Commun. 14(1), 145-159, 2022) when ℓ≠e/2.