Enumeration of extended irreducible binary Goppa codes

03/29/2022

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by   Bocong Chen, et al.

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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≥3 be a positive integer satisfying (r,n)=1 and (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.