Pure gaps on curves with many rational places

Abstract

We consider the algebraic curve defined by y m = f (x) where m ≥ 2 and f (x) is a rational function over F q . We extend the concept of pure gap to c-gap and obtain a criterion to decide when an s-tuple is a c-gap at s rational places on the curve. As an application, we obtain many families of pure gaps at two rational places on curves with many rational places.

Weierstrass pure gaps on curves with three distinguished points

“…Goppa codes supported on {P 1 , P 2 , P 3 } will be constructed, as follows. 2 Let us begin with Goppa codes supported on (P 1 , P 2 ). For i, j ≥ 1 and i + j = d ≤ n − 1, it follows from Theorem 1.1 that the pairs…”

“…The latter study was extended to several points by Carvalho and Torres [5]. After these two seminal papers, many authors have pursuit the characterization of Weierstrass semigroups and pure gaps on special families of curves [2], [4], [10], [15]. This problem, which involves determining the dimension of certain divisors, is challenging and important in its own right.…”

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.…”

“…Let q = 3. The 167 pure gaps are in the set G 0 (Q 1 , Q 2 ) given below:A ∪ A ∪ (1, 1), (2, 2),(3,3), (4, 4), (5, 5), (7, 7),(8,8),(11,11),(14,14) , 2), (1, 3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,10),(1,11),(1,13),(1,14),(1,15),(1,16),(1,17),(1,19),(1,20),(1,23), (2, 3), (2, 4), (2, 5), (2, 6), (2, 7), (2, 8), (2, 10), (2, 11), (2, 13),(2,14),(2,15),(2,16),(2,17),(2,19),(2,20),…”

“…1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23, 24, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 82, 83, 84, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 107, 108,109, 113, 114, 119, 121, 122, 123, 124, 127, 128, 129, 133, 134, 139, 152, 153, 154, 158, 159, 164, 183, 184, 189, 214 …”