Autocorrelations of Random Binary Sequences

Abstract

We define B n to be the set of n-tuples of the form (a 0 , . . . , a n−1 ) where a j = ±1. If A ∈ B n , then we call A a binary sequence and define the autocorrelations of A by c k := n−k−1 j=0 a j a j+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 b(n) := min A∈Bn max 1 k n−1 |c k |.The purpose of this article is to show that, using some known techniques from discrete probability, we can improve upon the best upper bound on b(n) appearing in the previous literature, and we can obtain both asymptotic and exact expressions for the expected value of c m k if the a j are independent ±1 random variables with mean 0. We also include some brief heuristic remarks in support of the unproved conjecture that b(n) = O( √ n).

Reducing pulse compression sidelobes by inversion-amenable code selection

“…This stance is in contrast to many of those seen in the quest for low correlated binary sequences [2], [3] with no follow-up filtering. The MPS codes explicitly evaluate the code ACF and the sidelobes encountered; this search, though burdensome, is the most dependable course of action.…”

“…Barker codes (which exist only for length N=13 and some smaller values) all have 20 log 10 (1/N) as their peak sidelobe level while non-Barker codes have higher values (and, hence, more prominent false target replicas at the output of their matched filters), only decreasing (non-monotonically) agonizingly slowly with N [3]. The search space is enormous, requiring 2 N evaluations for each chosen code length, so it is no trivial matter to examine and assess large codes.…”

Typical peak sidelobe level of binary sequences

“…Apparently the suggested approach also allows proving that this bound is indeed true for most of the sequences (see comments at the bottom of p.670 in [18]). Dmitriev and Jedwab [4] conjectured and provided an experimental evidence that the typical PSL behaves as Θ( √ n ln n).…”

“…Moon and Moser [19] proved that for almost all sequences, κ(n) ≤ M (S n ) ≤ (2 + ) √ n ln n, for any κ(n) = o( √ n). Mercer [18] showed that…”

Bounds on the growth rate of the peak sidelobe level of binary sequences

“…(It is clear from Theorem 1 (i) that for any fixed > 0, M n ≤ (2 + ) √ n log n when n is sufficiently large. The constant in this latter bound was improved from 2 to √ 2 by Mercer [8], but his proof applies only to M n and not to almost all binary sequences. )…”