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The Euclidean hull of a linear code $C$ is the intersection of $C$ with its Euclidean dual $C^\perp $. The hull with low dimensions gets much interest due to its crucial role in determining the complexity of algorithms for computing the automorphism group of a linear code and for checking permutation equivalence of two linear codes. The Euclidean hull of a linear code has been applied to the so-called entanglement-assisted quantum error-correcting codes (EAQECCs) via classical error-correcting codes. In this paper, we firstly consider linear codes with one-dimensional Euclidean hull from algebraic geometry codes, and then present a general method to construct linear codes with arbitrary dimensional Euclidean hull. Some new EAQECCs are presented. [ABSTRACT FROM AUTHOR]

Let $ \mathbb K$ be an algebraically closed field. In this paper, we consider the class of smooth plane curves of degree $n+1>3$ over $ \mathbb K$ , containing three points, $P_{1},P_{2}$ , and $P_{3}$ , such that $nP_{1}+P_{2}$ , $nP_{2}+P_{3}$ , and $nP_{3}+P_{1}$ are divisors cut out by three distinct lines. For such curves, we determine the dimension of certain special divisors supported on $\{P_{1},P_{2},P_{3}\}$ , as well as an explicit description of all pure gaps at each nonempty subset of the distinguished points $P_{1},P_{2}$ , and $P_{3}$. When $ \mathbb K=\overline { \mathbb F}_{q}$ , this class of curves, which includes the Hermitian curve, is used to construct algebraic geometry codes having minimum distance better than the Goppa bound. [ABSTRACT FROM AUTHOR]