One- and Two-Point Codes Over Kummer Extensions

Abstract

Abstract. We compute the Weierstrass semigroup at one totally ramified place for Kummer extensions defined by y m = f (x) λ where f (x) is a separable polynomial over F q . In addition,we compute the Weierstrass semigroup at two certain totally ramified places. We then apply our results to construct one-and two-point algebraic geometric codes with good parameters.

Pure Weierstrass gaps from a quotient of the Hermitian curve

“…Weierstrass semigroups and pure gaps are of significant uses in the construction and analysis of AG codes for their applications in obtaining codes with good parameters (see [8], [9]). In [3], [10], Garcia, Kim and Lax improved the Goppa bound using the arithmetical structure of the Weierstrass gaps at only one place.…”

“…In [15], Matthews generalized the results of [7], [14] by determining the Weierstrass semigroup of arbitrary rational places on the quotient of the Hermitian curve defined by the equation y m = x q + x over F q 2 where q is a prime power and m > 2 is a divisor of q + 1. For general Kummer extensions, the authors in [16], [9] recently described the Weierstrass semigroups and gaps at one place and two places. Bartoli, Quoos and Zini [17] gave a criterion to find pure gaps at many places and presented families of pure gaps.…”

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].…”

The root diagram for one-point AG codes arising from certain curves with separated variables

“…In this work, we will construct the root diagram, and consequently an algorithm for computing a Gröbner basis, for codes arising from certain curves over F q with automorphisms whose order divides q − 1, thus we get results more general than those achieved previously. As examples, we have codes over the curves y q + y = x q r +1 and y q + y = x m , and codes over Kummer extensions, which have been applied in coding theory, see [9] and [12], and [2], respectively. This paper is organized as follows.…”