  The problem of constructing polynomials with all coefficients $1$ or $-1$ and
large merit factor (equivalently with small $L^4$ 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.
