Enumeration of extended irreducible binary Goppa codes

TL;DR

This paper introduces a new method to estimate the upper bound on the number of inequivalent extended irreducible binary Goppa codes, which are crucial for cryptographic security assessments.

Contribution

It provides a novel approach to bound the count of inequivalent Goppa codes with specific parameters, aiding cryptosystem security evaluation.

Findings

Established an upper bound for the number of inequivalent codes

Focused on codes with length q+1 and degree r

Applicable for odd prime n and specific gcd conditions

Abstract

The family of Goppa codes is one of the most interesting subclasses of linear codes. As the McEliece cryptosystem often chooses a random Goppa code as its key,knowledge of the number of inequivalent Goppa codes for fixed parameters may facilitate in the evaluation of the security of such a cryptosystem. In this paper we present a new approach to give an upper bound on the number of inequivalent extended irreducible binary Goppa codes. To be more specific, let $n>3$ be an odd prime number and $q=2^n$; let $r\geq3$ be a positive integer satisfying $\gcd(r,n)=1$ and $\gcd\big(r,q(q^2-1)\big)=1$. We obtain an upper bound for the number of inequivalent extended irreducible binary Goppa codes of length $q+1$ and degree $r$.