A new construction of nonlinear codes via rational function fields

TL;DR

This paper introduces a novel explicit method for constructing nonlinear codes over (q+1)-ary alphabets using rational function fields, allowing evaluations at pole places, resulting in codes with near-optimal parameters.

Contribution

It presents a new construction of nonlinear codes via rational functions evaluated at all rational places, including poles, expanding the methods beyond traditional algebraic geometry codes.

Findings

Codes have parameters close to the Singleton bound.

Codes outperform those derived from MDS codes through alphabet restriction or extension.

Construction is explicit and applicable over any prime power q.

Abstract

It is well known that constructing codes with good parameters is one of the most important and fundamental problems in coding theory. Though a great many of good codes have been produced, most of them are defined over alphabets of sizes equal to prime powers. In this paper, we provide a new explicit construction of (q+1)-ary nonlinear codes via rational function fields, where q is a prime power. Our codes are constructed by evaluations of rational functions at all the rational places (including the place of "infinity") of the rational function field. Compared to the rational 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 with length q+1 over the alphabet $\mathbb{F}_{q}\cup \{\infty\}$. As a result, our codes have reasonable parameters as they are very close to the Singleton bound. Furthermore, our codes have better parameters than those obtained from MDS codes via code alphabet restriction or extension.