Enumeration of Extended Irreducible Binary Goppa Codes
Bocong Chen, Guanghui Zhang · IEEE Transactions on Information Theory · 2022
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$r\geq 3$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$r$.