On Galois hulls of linear codes and new entanglement-assisted quantum error-correcting codes

The Galois inner product is a generalization of the Euclidean inner product and Hermitian inner product. The Galois hull of a linear code is the intersection of itself and its Galois dual code, which has aroused the interest of researchers in these years. In this paper, we study Galois hulls of linear codes. Firstly, the symmetry of the dimensions of Galois hulls is found. Some new necessary and sufficient conditions for linear codes being Galois self-orthogonal codes, Galois self-dual codes and Galois linear complementary dual codes are characterized. Then, based on these properties, we develop the previous theory and propose explicit methods to construct Galois self-orthogonal codes of lengths $n+2i$ ($i\geq 0$) and $n+2i+1$ ($i\geq 1$) from Galois self-orthogonal codes of length $n$. As applications, linear codes of lengths $n+2i$ and $n+2i+1$ with Galois hulls of arbitrary dimensions are derived immediately. After this, two new classes of Hermitian self-orthogonal MDS codes are also constructed. Finally, applying all the results to the constructions of entanglement-assisted quantum error-correcting codes (EAQECCs), many new EAQECCs and MDS EAQECCs with rates greater than or equal to $\frac{1}{2}$ and positive net rates can be obtained.