Weierstrass semigroups of maximal function fields with applications to AG codes

In this thesis, we study Weierstrass semigroups at one or multiple places of certain maximal function fields, with a twofold purpose: extending the theoretical knowledge on these function fields and investigating applications to Algebraic Geometry codes (AG codes).

The first major contribution of the thesis consists in the determination of the Weierstrass semigroups at all the places of one of the known maximal function fields with the third largest genus. Consequently, we are also able to determine the full automorphism group of the function field. We find several different types of Weierstrass semigroups and a surprisingly rich set of Weierstrass places, which had never been observed before for any of the other maximal function fields for which the Weierstrass places are known.

A second major contribution presented in this dissertation is the computation of the Weierstrass semigroups at certain pairs of places of two different families of maximal function fields: the Beelen-Montanucci function fields and the Skabelund function field obtained as a cyclic extension of the Suzuki one. As a result, we are able to estimate the minimum distance of certain two-point AG codes from these function fields, obtaining improvements on comparable AG codes that had previously been studied in the literature.

As a final contribution, we present the study of upper and lower bounds for a constant that captures the asymptotic behaviour of the number of rational points of projective curves over a finite field, when the degree of the curve becomes large with respect to the field cardinality. The exact value of the constant remains unknown, but improvements to the previously known bounds are found.