The number of extended irreducible binary Goppa codes

04/05/2022
by   Bocong Chen, et al.
0

Goppa, in the 1970s, discovered the relation between algebraic geometry and codes, which led to the family of Goppa codes. As one of the most interesting subclasses of linear codes, the family of Goppa codes is often chosen as a key in the McEliece cryptosystem. Knowledge of the number of inequivalent binary Goppa codes for fixed parameters may facilitate in the evaluation of the security of such a cryptosystem. Let n≥5 be an odd prime number, let q=2^n and let r≥3 be a positive integer satisfying (r,n)=1. The purpose of this paper is to establish an upper bound on the number of inequivalent extended irreducible binary Goppa codes of length q+1 and degree r.A potential mathematical object for this purpose is to count the number of orbits of the projective semi-linear group PGL_2(𝔽_q)⋊ Gal(𝔽_q^r/𝔽_2) on the set ℐ_r of all monic irreducible polynomials of degree r over the finite field 𝔽_q. An explicit formula for the number of orbits of PGL_2(𝔽_q)⋊ Gal(𝔽_q^r/𝔽_2) on ℐ_r is given, and consequently, an upper bound for the number of inequivalent extended irreducible binary Goppa codes of length q+1 and degree r is derived. Our main result naturally contains the main results of Ryan (IEEE-TIT 2015), Huang and Yue (IEEE-TIT, 2022) and, Chen and Zhang (IEEE-TIT, 2022), which considered the cases r=4, r=6 and (r,q^3-q)=1 respectively.

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