Goppa codes are particularly appealing for cryptographic applications. Every improvement of our knowledge of Goppa codes is of particular interest. In this paper, we present a sufficient and necessary condition for an irreducible monic polynomial $g(x)$ of degree $r$ over $\mathbb{F}_{q}$ satisfying $\gamma g(x)=(x+d)^rg({A}(x))$, where $q=2^n$, $A=\left(\begin{array}{cc} a&b\\1&d\end{array}\right)\in PGL_2(\Bbb F_{q})$, $\mathrm{ord}(A)$ is a prime, $g(a)\ne 0$, and $0\ne \gamma\in \Bbb F_q$. And we give a complete characterization of irreducible polynomials $g(x)$ of degree $2s$ or $3s$ as above, where $s$ is a positive integer. Moreover, we construct some binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes.
翻译：Goppa 代码对于加密应用特别具有吸引力。 我们对Goppa 代码知识的每一项改进都特别感兴趣。 在本文中, 我们提出了一个足够和必要的条件, 用于一个不可复制的单倍多价$g(x) $g(x) $g(x) =(x+d) rg({A}}(x) $), 其中$q=2美元, $left(\gin{ray}cc} a&b\\\1&d\end{rary{right)\ a & bray}\ a & d\end{{right} in PGL_2(\2(\\\ bb F}q})$($\ mathrm{ord}(A) $(x) $(x) $(x) $@g(x) +d) =(x) =(x) g(x) =(x) +d) rg(x) $(x) $(x) $(x(x) left(x) $) $(x(x) $(x) $2s(x) $) $(x(x) as(x) $) $) $(x(x) $) $) $) $(x) a(x) as(x) $(x(x) ax(或3s(dendend) a) a $) a(或 $) a(或dend) a(dend) a(dend) a(dend) a(dend)\dend) a(或 $) a(dend) a(3s) a(或 $) $) $) $) a(或 $) a(或 $) a(x) in PGGGL_(或 $) $) in PGL_(或 $) $) a(或 $) $) $) $) $(或 $) $(或 $) $) $(x) $(x) $(
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