In this article we prove that a class of Goppa codes whose Goppa polynomial is of the form $g(x) = \Tr(x)$ (i.e. $g(x)$ is a trace polynomial from a field extension of degree $m \geq 3$) has a better minimum distance then 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.
翻译:在本篇文章中,我们证明,Goppa多元面值为$g(x)=\Tr(x)$(即$g(x)美元)的Goppa代码类别,其最低距离比Goppa约束$d\geq 2deg(g(x))+1美元的Goppa代码的最小距离要好。我们的改进是基于找到另一个Goppa多元面值$(L,g)=C(M,h)美元,但$deg(h) > deg(g)美元。这是与Trace Goppa代码相比,四方面值(即,案件$m=2美元)的显著改进,因为Goppa将四面体案捆绑在一起的Goppa代码是锐利的。