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 $\gamma g(x)=(x+d{)}^{r}g(A(x))$, where $q=2^n$, $A=\left(\begin{array}{cc}a& b\\ 1& d\end{array}\right)\in PG{L}_2({\mathbb{F}}_q)$, $\mathrm{ord}(A)$ is a prime, $g(a)\ne 0$, and $0\ne \gamma \in {\mathbb{F}}_q$. 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.