Enumeration of extended irreducible binary Goppa codes

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 $r\ge 3$ be a positive integer satisfying $\gcd (r,n)=1$ and $\gcd (r,q(q^2-1))=1$. We obtain an upper bound for the number of inequivalent extended irreducible binary Goppa codes of length $q+1$ and degree $r$.

This paper is already accepted by IEEE Transactions on Information Theory on March 3, 2022