Characterization of optimal binary linear codes with one-dimensional hull

11/04/2022

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by   Shitao Li, et al.

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The hull of a linear code over finite fields is the intersection of the code and its dual. Linear codes with small hulls have been widely studied due to their applications in computational complexity and information protection. In this paper, we study some properties of binary linear codes with one-dimensional hull, and establish their relation with binary LCD codes. Some interesting inequalities are thus obtained. We determine the exact value of d_one(n,k) for k∈{1,3,4,n-5,n-4,n-3,n-2,n-1} or 14≤ n≤ 24, where d_one(n,k) denotes the largest minimum weight among all binary linear [n,k] codes with one-dimensional hull. We partially determine the value of d_one(n,k) for k=5 or 25≤ n≤ 30. As an application, we construct some entanglement-assisted quantum error-correcting codes (EAQECCs).