A New Construction of Nonlinear Codes via Algebraic Function Fields

Abstract

In coding theory, constructing codes with good parameters is one of the most important and fundamental problems. A great many good codes have been constructed over alphabets of sizes equal to prime powers, however, good block codes over other alphabet sizes are rare. In this paper, we provide a new explicit construction of ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> + 1)-ary nonlinear codes via algebraic function fields, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> is a prime power. Our codes are constructed by evaluating rational functions at all rational places of an algebraic function field. Compared with algebraic geometry codes, the main difference is that we allow rational functions to be evaluated at pole places. After evaluating rational functions from a union of Riemann-Roch spaces, we obtain a family of nonlinear codes over the alphabet F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>q</i></sub> ⋃ {∞}. It turns out that our codes have better parameters than those obtained from MDS codes or good algebraic geometry codes via code alphabet extension and restriction.