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The problem of constructing polynomials with all coefficients $1$ or $-1$ andlarge merit factor (equivalently with small $L^4$ norm on the unit circle)arises naturally in complex analysis, condensed matter physics, and digitalcommunications engineering. Most known constructions arise (sometimes in asubtle way) from difference sets, in particular from Paley and Singerdifference sets. We consider the asymptotic merit factor of polynomialsconstructed from other difference sets, providing the first essentially newexamples since 1991. In particular we prove a general theorem on the asymptoticmerit factor of polynomials arising from cyclotomy, which includes results onHall and Paley difference sets as special cases. In addition, we establish theasymptotic merit factor of polynomials derived from Gordon-Mills-Welchdifference sets and Sidelnikov almost difference sets, proving two recentconjectures.

Merit factors of polynomials derived from difference sets