Extended Irreducible Binary Sextic Goppa Codes

Enumeration of extended irreducible binary Goppa codes

“…The primary purpose of this paper is to give an upper bound for the number of (inequivalent) extended irreducible binary Goppa codes of length q + 1 and degree r. This problem can be reduced to that of counting the number of orbits of the projective semi-linear group action on some set (see [1], [7] or [24]). To state this result clearly, we need the notions of group actions and some matrix groups.…”

“…The papers [7], [13], [17], [18] and [24] used the Cauchy-Frobenius Theorem to calculate the number of orbits of PΓL on S. Here we introduce another group action: The group PΓL can act on the set of all monic irreducible polynomials of degree r over F q . Let I r be the set of all monic irreducible polynomials of degree r over F q .…”

“…In [17], an upper bound on the number of inequivalent extended irreducible binary Goppa codes of degree 2p and length 2 n + 1 was produced, where n and p are two distinct odd primes such that p does not divide 2 n ± 1. Very recently, Huang and Yue [7] obtained an upper bound on the number of extended irreducible binary Goppa codes of degree 6 and length 2 n + 1, where n > 3 is a prime number.…”

“…In this paper, we explore further the ideas in [7] and [24] to give an upper bound on the number of inequivalent extended irreducible binary Goppa codes of length 2 n + 1 and degree r, where n > 3 is an odd prime number and r ≥ 3 be a positive integer satisfying gcd(r, n) = 1 and gcd r, q(q 2 − 1) = 1. By virtue of a result in [24], the number of inequivalent extended irreducible binary Goppa codes of length 2 n + 1 and degree r is less than or equal to the number of orbits of PΓL on S, where PΓL = PGL ⋊ Gal is the projective semi-linear group and S is a subset of F 2 nr .…”

“…By virtue of a result in [24], the number of inequivalent extended irreducible binary Goppa codes of length 2 n + 1 and degree r is less than or equal to the number of orbits of PΓL on S, where PΓL = PGL ⋊ Gal is the projective semi-linear group and S is a subset of F 2 nr . With the help of this result, one only needs to count the number of orbits of PΓL on S. The papers [7], [13], [17], [18] and [24] used the Cauchy-Frobenius Theorem to calculate the number of orbits of PΓL on S. Distinguishing from this approach, we do not use the Cauchy-Frobenius Theorem to obtain our main result; we introduce an action of PΓL on the set of all monic irreducible polynomials of degree r over F q , say I r (see Lemma 3.1). We then show that the number of orbits of PΓL on S is equal to the number of orbits of PΓL on I r (see Lemma 3.3).…”