Möbius–frobenius Maps on Irreducible Polynomials

Martínez, F.E. Brochero; Oliveira, Daniela; Reis, Lucas

doi:10.1017/s0004972720001306

Cited by 4 publications

(2 citation statements)

References 6 publications

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“…Many authors have studied the action of PGL on I r , focusing on the characterization and number of A-invariants where A ∈ PGL (for example, see [4], [19], [20], [21], [27]). The paper [14] considered an action of PΓL on I r , and our definition of PΓL on I r is different from that of [14].…”

Section: The Action Of Pγl On Imentioning

confidence: 99%

Enumeration of extended irreducible binary Goppa codes

Chen¹,

Zhang²

2022

Preprint

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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 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.

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Section: The Action Of Pγl On Imentioning

confidence: 99%

Enumeration of extended irreducible binary Goppa codes

Chen¹,

Zhang²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…Many authors have studied the action of PGL on I r , focusing on the characterization and number of A-invariants where A ∈ PGL (for example, see [5], [22], [23], [24], [29]). The paper [16] considered an action of PΓL on I r , and our definition of PΓL on I r is different from that of [16].…”

Section: The Action Of Pγl On I Rmentioning

confidence: 99%

The number of extended irreducible binary Goppa codes

Chen¹,

Zhang²

2022

Preprint

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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 gcd(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 semilinear group PGL2(Fq) ⋊ Gal(Fqr /F2) on the set Ir of all monic irreducible polynomials of degree r over the finite field Fq. An explicit formula for the number of orbits of PGL2(Fq) ⋊ Gal(Fqr /F2) on Ir 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 gcd(r, q 3 − q) = 1 respectively. MSC: 94B50.

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Enumeration of Extended Irreducible Binary Goppa Codes

Chen

Zhang

2022

IEEE Trans. Inform. Theory

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Möbius–frobenius Maps on Irreducible Polynomials

Cited by 4 publications

References 6 publications

Enumeration of extended irreducible binary Goppa codes

Enumeration of extended irreducible binary Goppa codes

The number of extended irreducible binary Goppa codes

Enumeration of Extended Irreducible Binary Goppa Codes

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