Weierstrass semigroup on special algebraic curves

An important example of numerical semigroups is the Weierstrass semigroup at a point on a curve. Weierstrass semigroups and gonality sequences have applications in many branches of mathematics such as algebraic geometry and coding theory. Weierstrass semigroups and gonality sequences, in particular, they play an important role in studies aimed at establishing new codes, finding limits for minimum distances, and making improvements in the size of codes. In this thesis; Hermitian and Suzuki function fields along with the basic concepts of the function field ̃ 𝑆 = 𝔽𝑞(𝑥, 𝑦, 𝑡)/𝔽𝑞, which is a Kummer extension of the Suzuki function field established by Skabelund (2018) as 𝑦𝑞 + 𝑦 = 𝑥𝑞0 (𝑥𝑞 + 𝑥), 𝑡𝑚 = 𝑥𝑞 + 𝑥, are given, rational points on Weierstrass semigroups are investigated and also recovered by using techniques from the work of Beelen vd. (2021), Bartoli vd. (2021). Here 𝑛 ≥ 1, 𝑞0 = 2𝑛, 𝑞 = 2𝑞2 0 , and 𝑚 = 𝑞 − 2𝑞0 + 1. In addition, the relationship between the gonality sequence in a function field and the Weierstrass semigroup is examined and the gonality sequence is given for some genus values.