Binary irreducible quasi-cyclic parity-check subcodes of Goppa codes and extended Goppa codes

Abstract: 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.