Merit factors of polynomials derived from difference sets

TL;DR

This paper investigates the asymptotic merit factors of polynomials derived from various difference sets, introducing new examples and proving conjectures related to cyclotomy, Gordon-Mills-Welch, and Sidelnikov difference sets.

Contribution

It provides the first new asymptotic merit factor results since 1991 for polynomials from difference sets, including a general theorem on cyclotomy and proofs of recent conjectures.

Findings

Established asymptotic merit factors for cyclotomy-based polynomials.

Proved merit factor values for Gordon-Mills-Welch and Sidelnikov difference sets.

Extended the class of difference sets known to produce polynomials with large merit factors.

Abstract

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.