Advances in the merit factor problem for binary sequences

TL;DR

This paper advances the understanding of the merit factor in binary sequences by establishing asymptotic values, proving conjectures, and introducing new families of sequences with high merit factors, relevant in communications and physics.

Contribution

It establishes the asymptotic merit factor for several binary sequence families, including skew-symmetric sequences, and unifies scattered results using Fourier and combinatorial methods.

Findings

Proved conjectures about merit factors of binary sequences.

Identified skew-symmetric sequences with optimal asymptotic merit factor.

Unified various results within a single analytical framework.

Abstract

The identification of binary sequences with large merit factor (small mean-squared aperiodic autocorrelation) is an old problem of complex analysis and combinatorial optimization, with practical importance in digital communications engineering and condensed matter physics. We establish the asymptotic merit factor of several families of binary sequences and thereby prove various conjectures, explain numerical evidence presented by other authors, and bring together within a single framework results previously appearing in scattered form. We exhibit, for the first time, families of skew-symmetric sequences whose asymptotic merit factor is as large as the best known value (an algebraic number greater than 6.34) for all binary sequences; this is interesting in light of Golay's conjecture that the subclass of skew-symmetric sequences has asymptotically optimal merit factor. Our methods combine Fourier analysis, estimation of character sums, and estimation of the number of lattice points in polyhedra.