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.
翻译:在本篇文章中,我们证明,Goppa多元面值为$g(x) = x + x qq +\ cdots + x Qq ⁇ m-1 $,其中美元= geq 3美元(即 $g(x) 美元)是来自度度延伸的微量多元值($m\ geq 3美元)的一组Goppa代码的最小距离比Goppa捆绑 $d geq 2deg(g(x))+1美元意味着的要好。我们的改进是基于找到另一个Goppa 多元面值美元,例如$C(g) = C(M), h) = $deg(h) > deg(g) $。这是对二次场扩展的Trace Goppa代码(即案件 $m = 2美元) 的重大改进,因为用于二次扩展的Goppa的Goppa代码非常。