Merit factors of polynomials derived from difference sets

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Merit factors of polynomials derived from difference sets

The problem of constructing polynomials with all coefficients 1 or −1 and large merit factor (equivalently with small L norm on the unit circle) arises naturally in complex analysis, condensed matter physics, and digital communications engineering. Most known constructions arise (sometimes in a subtle way) from difference sets, in particular from Paley and Singer difference sets. We consider th...

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Skew Hadamard Difference Sets from Dickson Polynomials of Order 7

Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in Fq where q ≡ 3 mod 4) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of D5(x2, u) is a new skew Hadamard...

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Difference Sets and Polynomials of Prime Variables

Abstract. Let ψ(x) be a polynomial with rational coefficients. Suppose that ψ has the positive leading coefficient and zero constant term. Let A be a set of positive integers with the positive upper density. Then there exist x, y ∈ A and a prime p such that x− y = ψ(p− 1). Furthermore, if P is a set of primes with the positive relative upper density, then there exist x, y ∈ P and a prime p such...

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The merit factor of binary sequences related to difference sets

Long binary sequences related to cyclic difference sets are investigated. Among all known constructions of cyclic difference sets we show that only sequences constructed from Hadamard difference sets can have an asymptotic nonzero merit factor. Maximal length shift register sequences, Legendre, and twin-prime sequences are all constructed from Hadamard difference sets. We prove that the asympto...

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ژورنال

عنوان ژورنال: Journal of Combinatorial Theory, Series A

سال: 2017

ISSN: 0097-3165

DOI: 10.1016/j.jcta.2016.08.006