Bases and applications of Riemann-Roch Spaces of Function Fields with Many Rational Places

Algebraic geometry codes are generalizations of Reed-Solomon codes, which are implemented in nearly all digital communication devices. In ground-breaking work, Tsfasman, Vladut, and Zink showed the existence of a sequence of algebraic geometry codes that exceed the Gilbert-Varshamov bound, which was previously thought unbeatable. More recently, it has been shown that multipoint algebraic geometry codes can outperform comparable one-point algebraic geometry codes. In both cases, it is desirable that these function fields have many rational places. The prototypical example of such a function field is the Hermitian function field which is maximal. In 2003, Geil produced a new family of function fields which contain the Hermitian function field as a special case. This family, known as the norm-trace function field, has the advantage that codes from it may be defined over alphabets of larger size. The main topic of this dissertation is function fields arising from linearized polynomials; these are a generalization of Geil\u27s norm-trace function field. We also consider applications of this function field to error-correcting codes and small-bias sets. Additionally, we study certain Riemann-Roch spaces and codes arising from the Suzuki function field. The Weierstrass semigroup of a place on a function field is an object of classical interest and is tied to the dimension of associated Riemann-Roch spaces. In this dissertation, we derive Weierstrass semigroups of m-tuples of places on the norm-trace function fields. In addition, we also discuss Weierstrass semigroups from finite graphs