Improved bounds on the peak sidelobe level of binary sequences

TL;DR

This paper improves the upper bounds on the peak sidelobe level of binary sequences, showing that almost all sequences have levels below a refined threshold involving logarithmic factors.

Contribution

The authors establish tighter upper bounds on the peak sidelobe level for almost all binary sequences, advancing understanding of their typical autocorrelation properties.

Findings

Almost all binary sequences have peak sidelobe level at most rom previous bounds.

A new upper bound involving or the peak sidelobe level is proven.

A slightly better bound applies to a positive proportion of sequences.

Abstract

Schmidt proved in 2014 that if $\varepsilon>0$, almost all binary sequences of length $n$ have peak sidelobe level between $(\sqrt{2}-\varepsilon)\sqrt{n\log n}$ and $(\sqrt{2}+\varepsilon)\sqrt{n\log n}$. Because of the small gap between his upper and lower bounds, it is difficult to find improved upper bounds that hold for almost all binary sequences. In this note, we prove that if $\varepsilon>0$, then almost all binary sequences of length $n$ have peak sidelobe level at most $\sqrt{2n(\log n-(1-\varepsilon)\log\log n)}$, and we provide a slightly better upper bound that holds for a positive proportion of binary sequences of length $n$.