Enumeration of irreducible and extended irreducible Goppa codes

TL;DR

This paper establishes upper bounds on the number of irreducible and extended irreducible Goppa codes over finite fields, based on polynomial degree and field size, contributing to coding theory enumeration problems.

Contribution

It provides new upper bounds for counting irreducible and extended irreducible Goppa codes over finite fields, advancing understanding of their enumeration.

Findings

Derived upper bounds for irreducible Goppa codes

Derived upper bounds for extended irreducible Goppa codes

Applicable to codes defined by polynomials of degree r over GF(p)

Abstract

We obtain upper bounds on the number of irreducible and extended irreducible Goppa codes over $GF(p)$ of length $q$ and $q+1$, respectively defined by polynomials of degree $r$, where $q=p^t$ and $r\geq 3$ is a positive integer.