  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.
