Asymptotically Good Nonlinear Codes From Algebraic Curves

By employing algebraic curves, we give some new asymptotic bounds for (q-1)-ary and (q+1)-ary codes, where q >; 2 is a prime power. In particular, our asymptotic bound for (q-1)-ary codes improves on the bound obtained directly from alphabet restriction given by Tafasman and Vlăduţ , [Th. 1.3.19], while our asymptotic bound for (q+1) -ary codes includes Elkies´ result for the square q case (STOC 01) (however, the idea in this paper is different from Elkies´ one). Our constructions of asymptotically good nonlinear codes are NOT the same as Goppa´s construction of algebraic geometry codes in the sense that we consider evaluation of functions at some pole points as well.