Binary optimal linear codes with various hull dimensions and entanglement-assisted QECCs

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 (EAQECCs) with the best known parameters.