Generator Polynomials of Cyclic Expurgated or Extended Goppa Codes

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 $\Bbb F_{q}$ 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 $A \in PGL_{2}(\Bbb F_{q})$ . Moreover, we provide some examples to support our findings.