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 Οƒ duals for ΟƒβˆˆSLAut(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 Οƒ duals. We also show that the dimension of Οƒ hull of a linear code can be determined by a generator matrix of it or its Οƒ dual. We give a characterization on Οƒ dual and Οƒ 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 Οƒ dual. As an application, we construct eleven families of q-ary MDS codes with new β„“-Galois hulls satisfying 2(e-β„“)| 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 β„“β‰ e/2.

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