A new construction of nonlinear codes via algebraic function fields

TL;DR

This paper introduces a novel explicit method for constructing nonlinear codes over alphabets of size q+1 using algebraic function fields, allowing evaluations at pole places and achieving better parameters than existing codes.

Contribution

The paper presents a new construction of nonlinear codes via algebraic function fields, extending evaluation to pole places and improving code parameters over previous methods.

Findings

Codes have better parameters than MDS and algebraic geometry codes.

Evaluation at pole places is a key innovation.

Constructed codes are over the alphabet F_q {} } }.

Abstract

In coding theory, constructing codes with good parameters is one of the most important and fundamental problems. 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 algebraic function fields, where $q$ is a prime power. Our codes are constructed by evaluations of rational functions at all rational places of the 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 $\mathbb{F}_{q}\cup \{\infty\}$. 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.