Binary optimal linear codes with various hull dimensions and entanglement-assisted QECC
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The hull of a linear code C is the intersection of C with its dual. To the best of our knowledge, there are very few constructions of binary linear codes with the hull dimension ≥ 2 except for self-orthogonal codes. We propose a building-up construction to obtain a plenty of binary [n+2, k+1] codes with hull dimension ℓ, ℓ +1, or ℓ +2 from a given binary [n,k] code with hull dimension ℓ. In particular, with respect to hull dimensions 1 and 2, we construct all binary optimal [n, k] codes of lengths up to 13. With respect to hull dimensions 3, 4, and 5, we construct all binary optimal [n,k] codes of lengths up to 12 and the best possible minimum distances of [13,k] codes for 3 ≤ k ≤ 10. As an application, we apply our binary optimal codes with a given hull dimension to construct several entanglement-assisted quantum error-correcting codes(EAQECC) with the best known parameters.
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