Generator polynomials of cyclic expurgated or extended Goppa codes

Abstract

Classical Goppa codes are a well-known class of codes with applications in code-based cryptography, which are a special case of alternant codes. Many papers are devoted to the search for Goppa codes with a cyclic extension or with a cyclic parity-check subcode. Let ${\mathbb{F}}_q$ be a finite field with $q=2^l$ elements, where $l$ is a positive integer. In this paper, we determine all the generator polynomials of cyclic expurgated or extended Goppa codes under some prescribed permutations induced by the projective general linear automorphism $A\in PG{L}_2({\mathbb{F}}_q)$. Moreover, we provide some examples to support our findings.