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

Abstract

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 which almost all places of degree one split completely. Then we introduce the notion of ``quasi-symmetry'' of functions for $K/L$, and demonstrate its use in producing extensions of $F/K$ in which all places of degree one split completely. Using these techniques, we are able to restrict the ramification either to one chosen rational place, or entirely to non-rational places. We then apply these methods to the related problem of building asymptotically good towers of function fields. We construct examples of towers of function fields in which all rational places split completely throughout the tower. We construct Abelian towers with this property also. Furthermore, all of the above are done explicitly, ie., we give generators for the extensions, and equations that they satisfy. We also construct an integral basis for a set of places in a tower of function fields meeting the Drinfeld-Vladut bound using the discriminant of the tower localized at each place. Thus we are able to obtain a basis for a collection of functions that contains the set of regular functions in this tower. Regular functions are of interest in the theory of error-correcting codes as they lead to an explicit description of the code associated to the tower by providing the code's generator matrix.