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Title: On the Asymptotic Behavior of the Ratio between the Numbers of Binary Primitive and Irreducible Polynomials
Authors: Borissov, Yuri
Ho Lee, Moon
Nikova, Svetla
Keywords: Finite Fields
Primitive and Irreducible Polynomials
Issue Date: 2008
Publisher: Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation: Serdica Journal of Computing, Vol. 2, No 3, (2008), 239p-248p
Abstract: In this paper, we study the ratio θ(n) = λ2 (n) / ψ2 (n), where λ2 (n) is the number of primitive polynomials and ψ2 (n) is the number of irreducible polynomials in GF (2)[x] of degree n. Let n = ∏ pi^ri, i=1,..l be the prime factorization of n. We show that, for fixed l and ri , θ(n) is close to 1 and θ(2n) is not less than 2/3 for sufficiently large primes pi . We also describe an infinite series of values ns such that θ(ns ) is strictly less than 1/2.
Description: This work was presented in part at the 8th International Conference on Finite Fields and Applications Fq^8 , Melbourne, Australia, 9-13 July, 2007.
ISSN: 1312-6555
Appears in Collections:Volume 2 Number 3

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