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 ≥ 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. MSC: 94B50.

The number of extended irreducible binary Goppa codes

“…In this paper, we further explore the ideas in [3] to establish an upper bound on the number of inequivalent extended irreducible binary Goppa codes of length q + 1 and degree r, where q = 2 n and n ≥ 5 is a prime number satisfying gcd(r, n) = 1. In a word, we settle a much more general case by dropping the assumption gcd(r, q 3 − q) = 1 in [3]; consequently, our main results in the current paper naturally contain the main results of [3], [7], [20], [21] and [27]. A potential mathematical object for this purpose is to count the number of orbits of PΓL = PGL 2 (F q ) ⋊ Gal(F q r /F 2 ) on S (see Lemma 2.3 in Section 2).…”

“…A potential mathematical object for this purpose is to count the number of orbits of PΓL = PGL 2 (F q ) ⋊ Gal(F q r /F 2 ) on S (see Lemma 2.3 in Section 2). We first use a strategy exhibited in [3] to count the number of orbits of PΓL 2 (F q ) on I r , where I r denotes the set of monic irreducible polynomials over F q of degree r (see Lemmas 2.4 and 2.6 in Section 2). By virtue of a result in [27], the number of inequivalent extended irreducible binary Goppa codes of length q + 1 and degree r is less than or equal to the number of orbits of PΓL 2 (F q ) on S. We finally determine the exact value of the number of orbits of PΓL on I r (or equivalently PΓL on S), see Theorem 4.21 in Section 4.…”

“…By virtue of a result in [27], the number of inequivalent extended irreducible binary Goppa codes of length q + 1 and degree r is less than or equal to the number of orbits of PΓL 2 (F q ) on S. We finally determine the exact value of the number of orbits of PΓL on I r (or equivalently PΓL on S), see Theorem 4.21 in Section 4. Comparing to [3], without the assumption gcd(r, q 3 − q) = 1, we have to get around several difficulties in connecting the orbits of PΓL on I r and that on S (see Lemmas 4.2-4.6 in Section 4) and establish some new results (see . The auxiliary results may be interested in their own right.…”

“…Let I r be the set of all monic irreducible polynomials of degree r over F q . It has been shown that the number of orbits of PΓL on S is equal to the number of orbits of PΓL on I r , see [3].…”

“…Note that the degrees of the Goppa codes mentioned above are small or have at most two prime divisors. Chen and Zhang [3] presented a new approach to calculate the number of orbits of the projective semi-linear group on S yielding an upper bound on the number of extended irreducible binary Goppa codes of degree r and length 2 n + 1, where n > 3 is a prime number with gcd(r, n) = 1 and gcd(r, q 3 − q) = 1. In particular, the degree r of the Goppa code considered in [3] can have arbitrary many prime divisors.…”