Bounding the Peak Sidelobe Level of Binary Sequences of All Lengths

Improving upon 2010 results of Alon et al., it was shown in 2014 by Schmidt that asymptotically, almost all binary sequences of length n have peak sidelobe level close to (2n log n)
^1/2
. One specific result of Alon et al. is that if we fix ε > 0, then almost all binary sequences of length n have peak sidelobe level at most (2n(log n - (1.5 - ε) log log n))
^1/2
, in the sense that the probability of not satisfying that bound approaches 0 as n approaches infinity. In this note, we prove that for all sequence lengths n > 1, there is a binary sequence of length n with peak sidelobe level at most (2n(log n - log log n + 0.862))
^1/2
.