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Three measures of pseudorandomness of finite binary sequences were introduced by Mauduit and Sárközy in 1997 and have been studied extensively since then: the normality measure, the well-distribution measure, and the correlation measure of order r. Our main result is that the correlation measure of order r for random binary sequences converges strongly, and so has a limiting distribution. This solves a problem due to Alon, Kohayakawa, Mauduit, Moreira, and Rödl. We also show that the best known lower bounds for the minimum values of the correlation measures are simple consequences of a celebrated result due to Welch concerning the maximum nontrivial scalar products over a set of vectors. [ABSTRACT FROM AUTHOR]

A Barker sequence is a binary sequence for which all nontrivial aperiodic autocorrelations are at most 1 in magnitude. An old conjecture due to Turyn asserts that there is no Barker sequence of length greater than 13. In 1961, Turyn and Storer gave an elementary, though somewhat complicated, proof that this conjecture holds for odd lengths. We give a new and simpler proof of this result. [ABSTRACT FROM AUTHOR]

The extent to which a sequence of finite length differs from a shifted version of itself is measured by its aperiodic autocorrelations. Of particular interest are sequences whose entries are 1 or $$-1$$ , called binary sequences, and sequences whose entries are complex numbers of unit magnitude, called unimodular sequences. Since the 1950s, there is sustained interest in sequences with small aperiodic autocorrelations relative to the sequence length. One of the main motivations is that a sequence with small aperiodic autocorrelations is intrinsically suited for the separation of signals from noise, and therefore has natural applications in digital communications. This survey reviews the state of knowledge concerning the two central problems in this area: How small can the aperiodic autocorrelations of a binary or a unimodular sequence collectively be and how can we efficiently find the best such sequences? Since the analysis and construction of sequences with small aperiodic autocorrelations is closely tied to the (often much easier) analysis of periodic autocorrelation properties, several fundamental results on corresponding problems in the periodic setting are also reviewed. [ABSTRACT FROM AUTHOR]

Let $A_n=(a_0,a_1,\ldots ,a_{n-1})$ be drawn uniformly at random from ${\{ }-1,+1{\} }^n$ and define\[M(A_n)=\max _{01}{\mathrm {}}.\]It is proved that $M(A_n)/\sqrt {n\log n}$ converges in probability to $\sqrt {2}$. This settles a problem first studied by Moon and Moser in the 1960s and proves in the affirmative a recent conjecture due to Alon, Litsyn, and Shpunt. It is also shown that the expectation of $M(A_n)/\sqrt {n\log n}$ tends to $\sqrt {2}$. [ABSTRACT FROM AUTHOR]

Abstract: The identification of binary sequences with large merit factor (small mean-squared aperiodic autocorrelation) is an old problem of complex analysis and combinatorial optimization, with practical importance in digital communications engineering and condensed matter physics. We establish the asymptotic merit factor of several families of binary sequences and thereby prove various conjectures, explain numerical evidence presented by other authors, and bring together within a single framework results previously appearing in scattered form. We exhibit, for the first time, families of skew-symmetric sequences whose asymptotic merit factor is as large as the best known value (an algebraic number greater than 6.34) for all binary sequences; this is interesting in light of Golayʼs conjecture that the subclass of skew-symmetric sequences has asymptotically optimal merit factor. Our methods combine Fourier analysis, estimation of character sums, and estimation of the number of lattice points in polyhedra. [Copyright &y& Elsevier]