Upper bound on the number of inequivalent extended binary irreducible Goppa codes

Abstract

In this paper, we always assume that p and n>3 are two distinct odd primes, m≥2 is a positive integer, and gcd⁡(pm,2n(22n−1))≠1. Set r=p2 or r=p3, we determine upper bounds on the number of inequivalent extended binary irreducible Goppa codes Γ(L,g) of degree r and length 2n+1, where L=F2n∪{∞} is the support set and each g(x) is an irreducible polynomial of degree r over F2n.