Improving the minimum distance bound of Trace Goppa codes

TL;DR

This paper demonstrates that a specific class of Trace Goppa codes with polynomial $g(x)$ of degree $m \\geq 3$ have a minimum distance exceeding the classical Goppa bound, through finding alternative polynomials $h$ with higher degree.

Contribution

It introduces a method to improve the minimum distance bounds of Trace Goppa codes for degrees $m \\geq 3$ by identifying equivalent codes with higher degree polynomials.

Findings

Improved minimum distance bounds for Trace Goppa codes with $m \\geq 3$

Existence of higher degree polynomials $h$ representing the same code as $g$

Significant enhancement over the quadratic case where the Goppa bound is sharp

Abstract

In this article we prove that a class of Goppa codes whose Goppa polynomial is of the form $g(x) = x + x^q + \cdots + x^{q^{m-1}}$ where $m \geq 3$ (i.e. $g(x)$ is a trace polynomial from a field extension of degree $m \geq 3$) has a better minimum distance than what the Goppa bound $d \geq 2deg(g(x))+1$ implies. Our improvement is based on finding another Goppa polynomial $h$ such that $C(L,g) = C(M, h)$ but $deg(h) > deg(g)$. This is a significant improvement over Trace Goppa codes over quadratic field extensions (i.e. the case $m = 2$), as the Goppa bound for the quadratic case is sharp.