Linear codes of larger lengths with Galois hulls of arbitrary dimensions and related entanglement-assisted quantum error-correcting codes

Abstract

Galois hulls are generalizations of Euclidean and Hermitian hulls, which have attracted interest because of their important applications in determining the complexity of some algorithms about linear codes and in constructing entanglement-assisted quantum error-correcting codes (EAQECCs). In this paper, the whole contribution is two-folded. On one hand, we propose explicit methods to construct linear codes of larger lengths with Galois hulls of arbitrary dimensions from given Galois self-orthogonal codes. Conditions required in our approach are proven to be relatively weak. On the other hand, we apply these results to construct EAQECCs. Two bounds for EAQECCs constructed from linear codes with prescribed dimensional Galois hull are given and EAQECCs with rates greater than or equal to 12 and positive net rates can be obtained. We also present many interesting examples to explain visually how these two aspects work.