Merit factors of polynomials derived from difference sets

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• 作者：Christian Gü ; nther ^{chriguen@math.upb.de" class="auth_mail" title="E-mail the corresponding author} ; Kai-Uwe Schmidt ^{kus@math.upb.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Difference set ; Binary sequence ; Merit factor ; Asymptotic ; Character sum ; Aperiodic autocorrelation
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：145
• 期：Complete
• 页码：340-363
• 全文大小：471 K

文摘

The problem of constructing polynomials with all coefficients 1 or −1 and large merit factor (equivalently with small L⁴$L^4$ 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 the asymptotic merit factor of polynomials constructed from other difference sets, providing the first essentially new examples since 1991. In particular we prove a general theorem on the asymptotic merit factor of polynomials arising from cyclotomy, which includes results on Hall and Paley difference sets as special cases. In addition, we establish the asymptotic merit factor of polynomials derived from Gordon–Mills–Welch difference sets and Sidelnikov almost difference sets, proving two recent conjectures.