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Weierstrass Pure Gaps on Curves With Three Distinguished Points

2022; Institute of Electrical and Electronics Engineers; Volume: 68; Issue: 5 Linguagem: Inglês

10.1109/tit.2021.3140195

1557-9654

Herivelto Martins Borges Filho, Gregory Duran Cunha,

Commutative Algebra and Its Applications

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.