Excellent nonlinear codes from algebraic function fields

The Gilbert-Varshamov (GV) bound for asymptotic families of codes over F_q has been improved by Tsfasman, Vla˘dut$80, and Zink (TVZ) in 1982, and only recently further improvements have been obtained by Xing, Elkies, and Niederreiter-Özbudak, by considering also nonlinear codes. These improvements involve higher derivations in function fields and are very computational. We give in this correspondence a much simpler proof for those improvements. Our construction of asymptotically good nonlinear codes is very similar to Goppa´s construction of algebraic-geometry codes.