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The problem of constructing polynomials with all coefficients 1 or −1 and large merit factor (equivalently with small L 4 norm on the unit circle) arises naturally in complex analysis, condensed matter physics, and digital communications engineering. Most known constructions arise (sometimes in a subtle way) from difference sets, in particular from Paley and Singer difference sets. We consider the asymptotic merit factor of polynomials constructed from other difference sets, providing the first essentially new examples since 1991. In particular we prove a general theorem on the asymptotic merit factor of polynomials arising from cyclotomy, which includes results on Hall and Paley difference sets as special cases. In addition, we establish the asymptotic merit factor of polynomials derived from Gordon–Mills–Welch difference sets and Sidelnikov almost difference sets, proving two recent conjectures. [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]

A 3-phase Barker array is a matrix of third roots of unity for which all out-of-phase aperiodic autocorrelations have magnitude 0 or 1. The only known truly two-dimensional 3-phase Barker arrays have size 2 × 2 or 3 × 3. We use a mixture of combinatorial arguments and algebraic number theory to establish severe restrictions on the size of a 3-phase Barker array when at least one of its dimensions is divisible by 3. In particular, there exists a double-exponentially growing arithmetic function T such that no 3-phase Barker array of size [ABSTRACT FROM AUTHOR]

A binary sequence of length n is an n-tuple with elements in \-1,1\ and its peak sidelobe level is the largest absolute value of its aperiodic autocorrelations at nonzero shifts. A classical problem is to find binary sequences whose peak sidelobe level is small compared to the length of the sequence. Using known techniques from probabilistic combinatorics, this paper gives a construction for a binary sequence of length n with peak sidelobe level at most \sqrt2n\log (2n) for every n > 1. This improves the best known bound for the peak sidelobe level of a family of explicitly constructed binary sequences, which arises for the family of m-sequences. By numerical analysis, it is argued that the peak sidelobe level of the constructed sequences grows in fact like order \sqrtn\log\log n and, therefore, grows strictly more slowly than the peak sidelobe level of a typical binary sequence. [ABSTRACT FROM AUTHOR]

For integer r satisfying 0\leq r\leq p-2, a sequence family \Omega_r of polyphase sequences of prime period p, size (p-2)p^r, and maximum correlation at most 2+(r+1)\sqrtp is presented. The sequence families are nested, that is, \Omega_r is contained in \Omegar+1, which provides design flexibility with respect to family size and maximum correlation. The sequences in \Omega_r are derived from a combination of multiplicative and additive characters of a prime field. Estimates on hybrid character sums are then used to bound the maximum correlation. This construction generalizes \Omega_0, which was previously proposed by Scholtz and Welch. Sequence family \Omega_2 is closely related to a recent design by Wang and Gong, who bounded its maximum correlation using methods from representation theory and asked for a more direct proof of this bound. Such a proof is given here and an improvement of the bound is provided. [ABSTRACT FROM AUTHOR]