论文标题
改善跟踪GOPPA代码的最小距离结合
Improving the minimum distance bound of Trace Goppa codes
论文作者
论文摘要
在本文中，我们证明了一类GOPPA代码，其GOPPA多项式为$ g（x）= x + x + x^q + \ cdots + cdots + x^{q^{m-1} $ $ m \ geq 3 $（即$ g（x）$与trace that the the the the the geq 3 $ geq 3 $ \ geq 3 $ \ geq 3 $ \ geq 3 $ \ geq 3 $ \ ge。 $ d \ geq 2deg（g（x））+1 $暗示。我们的改进是基于找到另一个GOPPA多项式$ h $，以便$ c（l，g）= c（m，h）$，但$ deg（h）> deg（g）$。这是对二次场扩展的跟踪GOPPA代码的重大改进（即，$ M = 2 $），因为GOPPA绑定了二次情况下的情况。
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.
