Galois hulls of linear codes and new EAQECCs of arbitrary lengths

Let $q=p^h$ be a prime power and $e$ be an integer with $0\leq e\leq h-1$. The $e$-Galois hull of a linear code $\mathcal{C}$ is the intersection of $\mathcal{C}$ and its $e$-Galois dual code $\mathcal{C}^{\bot_e}$, which is of great significance in both theory and practice. In this paper, we first study the dimensions of Galois hulls of linear codes and find that the dimensions have symmetry. We further derive some new sufficient and necessary conditions for a linear code being an $e$-Galois self-orthogonal or linear complementary dual code based on this property. Then, we generalize the previous conclusions and propose the explicit constructions of $e$-Galois self-orthogonal codes of lengths $n+2i$ ($i\geq 0$) and $n+2i+1$ ($i\geq 1$) from $e$-Galois self-orthogonal codes of length $n$. It means that linear codes of lengths $n+2i$ and $n+2i+1$ with Galois hulls of arbitrary dimensions exist if an $e$-Galois self-orthogonal code of length $n$ exists. Finally, we construct two new classes of Hermitian self-orthogonal GRS codes and apply them together with all the results to the constructions of entanglement-assisted quantum error-correcting codes (EAQECCs for short). Some new EAQECCs and MDS EAQECCs of lengths $n+2i$ and $n+2i+1$ are obtained.