Transitive and Self-dual Codes Attaining the Tsfasman-vladut-zink Bound

A major problem in coding theory is the question if the class of cyclic codes is asymptotically good. In this paper we introduce as a generalization of cyclic codes the notion of transitive codes (see Definition 1.4 in Section 1), and we show that the class of transitive codes is asymptotically good. Even more, transitive codes attain the Tsfasman-Vladut-Zink bound over Fq, for all squares q = l . We also show that self-orthogonal and self-dual codes attain the Tsfasman-Vladut-Zink bound, thus improving previous results about self-dual codes attaining the Gilbert-Varshamov bound. The main tool is a new asymptotically optimal tower E0 ⊆ E1 ⊆ E2 ⊆ . . . of function fields over Fq (with q = l ), where all extensions En/E0 are Galois.