Counting Extended Irreducible Goppa Codes

TL;DR

This paper establishes an upper bound on the number of extended irreducible Goppa codes over finite fields, contributing to the understanding of their combinatorial properties and potential applications in coding theory.

Contribution

It provides a new upper bound for the count of extended irreducible Goppa codes of specified degree and length, advancing theoretical knowledge in coding theory.

Findings

Derived an explicit upper bound for code count

Focused on codes with prime degree and length parameters

Enhances understanding of code enumeration in finite fields

Abstract

We obtain an upper bound on the number of extended irreducible q-ary Goppa codes of degree $r$ and length $q^n+1$, where $q=p^t$ and $n$ and $r>2$ are prime numbers.