Weierstrass semigroups from Kummer extensions

Abstract

The Weierstrass semigroups and pure gaps can be helpful in constructing codes with better parameters. In this paper, we investigate explicitly the minimal generating set of the Weierstrass semigroups associated with several totally ramified places over arbitrary Kummer extensions. Applying the techniques provided by Matthews in her previous work, we extend the results of specific Kummer extensions studied in the literature. Some examples are included to illustrate our results.

Weierstrass Semigroups From a Tower of Function Fields Attaining the Drinfeld-Vladut Bound

“…It should be noted that Weierstrass semigroups and pure gaps are of vital use in finding AG codes with good parameters. We refer the reader to the works [3,4,5,13,14,16,18,22,23] for the technique to improve the parameters of AG codes using Weierstrass semigroups and pure gaps.…”

Pure Weierstrass gaps from a quotient of the Hermitian curve

“…Bartoli, Quoos and Zini [17] gave a criterion to find pure gaps at many places and presented families of pure gaps. In [18], [8], Hu and Yang explicitly determined the Weierstrass semigroups and pure gaps at many places, and constructed AG codes with excellent parameters. However, little is known about the numbers of gaps and pure gaps from algebraic curves.…”

Pure gaps on curves with many rational places

“…The majority of maximal curves and curves with many rational places has a plane model of Kummer-type. For those curves with affine equation given by y m = f (x) λ where m ≥ 2, λ ≥ 1 and f (x) is a separable polynomial over F q , general results on gaps and pure gaps can be found in [1,5,18,29]. Applications to codes on particular curves such as the Giulietti-Korchmáros curve, the Garcia-Güneri-Stichtenoth curve, and quotients of the Hermitian curve can be found in [19,28,31].…”