Weierstrass semigroups, pure gaps and codes on function fields.

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. Furthermore, we determine the Weierstrass semigroup at one and two totally ramified places in a Kummer extension defined by the affine equation y m = ∏ i = 1 r (x - α i) λ i over K, the algebraic closure of F q , where α 1 , ⋯ , α r ∈ K are pairwise distinct elements, 1 ≤ λ i < m , and gcd (m , ∑ i = 1 r λ i) = 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. [ABSTRACT FROM AUTHOR]