Fields, towers of function fields meeting asymptotic bounds, and basis constructions for algebraic-geometric codes

摘要

In this work, we use the notion of ``symmetry'' of functions for an extension$K/L$ of finite fields to produce extensions of a function field $F/K$ in whichalmost all places of degree one split completely. Then we introduce the notionof ``quasi-symmetry'' of functions for $K/L$, and demonstrate its use inproducing extensions of $F/K$ in which all places of degree one splitcompletely. Using these techniques, we are able to restrict the ramificationeither to one chosen rational place, or entirely to non-rational places. Wethen apply these methods to the related problem of building asymptotically goodtowers of function fields. We construct examples of towers of function fieldsin which all rational places split completely throughout the tower. Weconstruct Abelian towers with this property also. Furthermore, all of the above are done explicitly, ie., we give generatorsfor the extensions, and equations that they satisfy. We also construct an integral basis for a set of places in a tower offunction fields meeting the Drinfeld-Vladut bound using the discriminant of thetower localized at each place. Thus we are able to obtain a basis for acollection of functions that contains the set of regular functions in thistower. Regular functions are of interest in the theory of error-correctingcodes as they lead to an explicit description of the code associated to thetower by providing the code's generator matrix.