Nice Codes from Nice Curves

摘要

The well-known Tsfasman-Vladut-Zink (TVZ) theorem states that for all prime powers q = l{sup}2 ≥ 49 there exist sequences of linear codes over F{sub}q with increasing length whose limit parameters R and δ (rate and relative minimum distance) are better than the Gilbert-Varshamov bound. The basic ingredients in the proof of the TVZ theorem are sequences of modular curves (or their corresponding function fields) having many rational points in comparison to their genus (more precisely, these curves attain the so-called Drinfeld-Vladut bound). Starting with such a sequence of curves and using Goppa's construction of algebraic geometry (AG) codes, one easily obtains sequences of linear codes whose limit parameters beat the Gilbert-Varshamov bound. However, this construction yields just linear codes, and the question arises if one can refine the construction to obtain good long codes with additional nice properties (e.g., codes with many automorphisms, self-orthogonal codes or self-dual codes). This can be done. We give a brief outline of some results in this direction.