Extended Irreducible Binary Sextic Goppa codes

TL;DR

This paper establishes an upper bound on the number of extended irreducible binary Goppa codes of degree 6 over finite fields, contributing to the understanding of their enumeration and properties.

Contribution

It provides a new upper bound on the count of extended irreducible binary Goppa codes of degree 6, advancing coding theory knowledge.

Findings

Derived an upper bound for the number of such codes

Enhanced understanding of the structure of Goppa codes

Contributed to the enumeration of irreducible Goppa codes

Abstract

Let $n (>3)$ be a prime number and $\Bbb F_{2^n}$ a finite field of $2^n$ elements. Let $L =\Bbb F_{2^n}\cup \{\infty\}$ be the support set and $g(x)$ an irreducible polynomial of degree $6$ over $\Bbb F_{2^n}$. In this paper, we obtain an upper bound on the number of extended irreducible binary Goppa codes $\Gamma(L, g)$ of degree $6$ and length $2^n+1$.