Agradecimentos: Alonso S. Castellanos was partially supported by FAPEMIG: APQ 00696-18 and RED 0013-21. Erik A. R. Mendoza was partially supported by FAPERJ Grant 201.650/2021 and FAPESP Grant 2022/16369-2. Luciane Quoos thanks FAPERJ 260003/001703/2021 - APQ1, CNPQ PQ 302727/2019-1 and CAPES MATH AMSUD 88881.647739/2021-01 for the partial support Ver menos

Abstract: For an arbitrary function field, from the knowledge of the minimal generating set of the Weierstrass semigroup at two rational places, the set of pure gaps is characterized. Further-more, we determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation ym =r i=1(x . ƒ¿i )ƒÉi over K, the alge-
braic closure of Fq, where ƒ¿1, . . . , ƒ¿r ¸ K are pairwise distinct elements, 1 . ƒÉi < m, and gcd(m, ri=1 ƒÉi ) = 1. We apply these results to construct algebraic geometry codes over certain function fields with many rational places. For one-point codes we obtain families of codes with exact parameters Ver menos