Generator polynomials of cyclic expurgated or extended Goppa codes

TL;DR

This paper characterizes all generator polynomials of cyclic expurgated or extended Goppa codes under specific automorphisms, advancing understanding of their algebraic structure for cryptographic applications.

Contribution

It determines all generator polynomials of certain cyclic Goppa codes under specific automorphisms, providing explicit examples and expanding the algebraic understanding of these codes.

Findings

All generator polynomials of the specified codes are characterized.

Explicit examples illustrating the theoretical results are provided.

The results facilitate the design of codes with desired cyclic properties.

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 $\Bbb F_q$ 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.