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.

Chu sequences are a family of polyphase sequences that have perfect periodic autocorrelations and good aperiodic autocorrelations. It has previously been proved that the maximum offpeak (aperiodic) autocorrelation (in absolute value) of the Chu sequence of length n is asymptotically equal to 0.480261√n. It has also been empirically observed that the merit factor of Chu sequences appears to grow like a constant times √n. In this note, we provide an analytic proof that the merit factor of the Chu sequence of length n is bounded below by a constant multiple of √n for all n. To the author's knowledge, this is the first time a family of polyphase sequences of all lengths has been proved to have merit factor growing at least like order √n.