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 Vladut [12, Th. 1.3.19], while our asymptotic bound for (q+1)-ary codes includes Elkies' result for the square 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.
asymptotic bound goppa's construction algebraic geometry codes gilbert-varshamov bound