On the construction of tame towers over finite fields

Recently, W.-C. W. Li, et al. (Lect. Notes in Comput. Sci. 2369 (2002) 372) developed a non-deterministic algorithm to perform a computer search for polynomials that recursively define asymptotically good sequences of function fields. In this paper, we build on this work by refining this algorithm. We give many sufficient conditions for the construction of such sequences and we describe the techniques used in the search. Many examples are given. The resulting towers are important for the construction of asymptotically good sequences of codes and they could provide further numerical evidence for Elkies' modularity conjecture.