Improving the minimum distance bound of Trace Goppa codes

Abstract

In this paper we prove that the class of Goppa codes whose Goppa polynomial is of the form $g(x) = \textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}$ where $\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}$ is a trace polynomial from a field extension of degree $m \ge 3$ has a better minimum distance than what the Goppa bound $d \ge 2\deg (g(x))+1$ implies. This result is a significant improvement compared to the minimum distance of Trace Goppa codes over quadratic field extensions (the case $m = 2$). We present two different techniques to improve the minimum distance bound. For general p we prove that the Goppa code $C(L, \textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}})$ is equivalent to another Goppa code C(M, h) where $\deg (h) > \deg (\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}})$. For $p=2$ we use the fact that the values of $\textbf{Tr}_{{\mathbb {F}}_{q^{m}} \setminus {\mathbb {F}}_{q}}$ are fixed under q–powers to find several new parity check equations which increase the known distance bounds.