Autocorrelations of Random Binary Sequences

We define Bn to be the set of n-tuples of the form (a0, . . . , an−1) where aj = ±1. If A ∈ Bn, then we call A a binary sequence and define the autocorrelations of A by ck := ∑n−k−1 j=0 ajaj+k for 0 k n− 1. The problem of finding binary sequences with autocorrelations ‘near zero’ has arisen in communications engineering and is also relevant to conjectures of Littlewood and Erdős on ‘flat’ polynomials with ±1 coefficients. Following Turyn, we define