Advances in the merit factor problem for binary sequences

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 Golay's 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.Comment: 31 pages, minor revision

Low-Autocorrelation Binary Sequences: On Improved Merit Factors and Runtime Predictions to Achieve Them

The search for binary sequences with a high figure of merit, known as the low autocorrelation binary sequence ($labs$}) problem, represents a formidable computational challenge. To mitigate the computational constraints of the problem, we consider solvers that accept odd values of sequence length $L$ and return solutions for skew-symmetric binary sequences only -- with the consequence that not all best solutions under this constraint will be optimal for each $L$. In order to improve both, the search for best merit factor $and$ the asymptotic runtime performance, we instrumented three stochastic solvers, the first two are state-of-the-art solvers that rely on variants of memetic and tabu search ($lssMAts$ and $lssRRts$), the third solver ($lssOrel$) organizes the search as a sequence of independent contiguous self-avoiding walk segments. By adapting a rigorous statistical methodology to performance testing of all three combinatorial solvers, experiments show that the solver with the best asymptotic average-case performance, $lssOrel\_8 = 0.000032*1.1504^L$, has the best chance of finding solutions that improve, as $L$ increases, figures of merit reported to date. The same methodology can be applied to engineering new $labs$ solvers that may return merit factors even closer to the conjectured asymptotic value of 12.3248

Low Autocorrelation Binary Sequences: Number Theory-based Analysis for Minimum Energy Level, Barker codes

Low autocorrelation binary sequences (LABS) are very important for communication applications. And it is a notoriously difficult computational problem to find binary sequences with low aperiodic autocorrelations. The problem can also be stated in terms of finding binary sequences with minimum energy levels or maximum merit factor defined by M.J.E. Golay, F=N^2/2E, N and E being the sequence length and energy respectively. Conjectured asymptotic value of F is 12.32 for very long sequences. In this paper, a theorem has been proved to show that there are finite number of possible energy levels, spaced at an equal interval of 4, for the binary sequence of a particular length. Two more theorems are proved to derive the theoretical minimum energy level of a binary sequence of even and odd length of N to be N/2, and N-1/2 respectively, making the merit factor equal to N and N^2/N-1 respectively. The derived theoretical minimum energy level successfully explains the case of N =13, for which the merit factor (F =14.083) is higher than the conjectured value. Sequence of lengths 4, 5, 7, 11, 13 are also found to be following the theoretical minimum energy level. These sequences are exactly the Barker sequences which are widely used in direct-sequence spread spectrum and pulse compression radar systems because of their low autocorrelation properties. Further analysis shows physical reasoning in support of the conjecture that Barker sequences exists only when N <= 13 (this has been proven for all odd N).Comment: 34 page

Low Autocorrelation Binary Sequences

Binary sequences with minimal autocorrelations have applications in communication engineering, mathematics and computer science. In statistical physics they appear as groundstates of the Bernasconi model. Finding these sequences is a notoriously hard problem, that so far can be solved only by exhaustive search. We review recent algorithms and present a new algorithm that finds optimal sequences of length $N$ in time $\Theta(N\,1.73^N)$. We computed all optimal sequences for $N\leq 66$ and all optimal skewsymmetric sequences for $N\leq 119$.Comment: 17 pages, 4 figure

Note on the Merit Factors of Sequences

A method for estimating the merit factors of sequences will be provided. The result is also effective in determining the nonexistence of certain infinite collections of cyclic difference sets and cyclic matrices and associated binary sequences.Comment: Key Words: Binary sequences, Merit factor, Barker sequence, cyclic Hadamard matrix, Menon difference sets. 19 page

Sequences with Low Correlation

Pseudorandom sequences are used extensively in communications and remote sensing. Correlation provides one measure of pseudorandomness, and low correlation is an important factor determining the performance of digital sequences in applications. We consider the problem of constructing pairs $(f,g)$ of sequences such that both $f$ and $g$ have low mean square autocorrelation and $f$ and $g$ have low mean square mutual crosscorrelation. We focus on aperiodic correlation of binary sequences, and review recent contributions along with some historical context.Comment: 24 page

Improved Branch-and-Bound for Low Autocorrelation Binary Sequences

The Low Autocorrelation Binary Sequence problem has applications in telecommunications, is of theoretical interest to physicists, and has inspired many optimisation researchers. Metaheuristics for the problem have progressed greatly in recent years but complete search has not progressed since a branch-and-bound method of 1996. In this paper we find four ways of improving branch-and-bound, leading to a tighter relaxation, faster convergence to optimality, and better empirical scalability.Comment: Journal paper in preparatio

Aperiodic Crosscorrelation of Sequences Derived from Characters

It is shown that pairs of maximal linear recursive sequences (m-sequences) typically have mean square aperiodic crosscorrelation on par with that of random sequences, but that if one takes a pair of m-sequences where one is the reverse of the other, and shifts them appropriately, one can get significantly lower mean square aperiodic crosscorrelation. Sequence pairs with even lower mean square aperiodic crosscorrelation are constructed by taking a Legendre sequence, cyclically shifting it, and then cutting it (approximately) in half and using the halves as the sequences of the pair. In some of these constructions, the mean square aperiodic crosscorrelation can be lowered further if one truncates or periodically extends (appends) the sequences. Exact asymptotic formulae for mean squared aperiodic crosscorrelation are proved for sequences derived from additive characters (including m-sequences and modified versions thereof) and multiplicative characters (including Legendre sequences and their relatives). Data is presented that shows that sequences of modest length have performance that closely approximates the asymptotic formulae.Comment: 54 page

Merit factors of polynomials derived from difference sets

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.Comment: 22 pages, this revision contains a more general version of Thm. 2.

Low Correlation Sequences from Linear Combinations of Characters

Pairs of binary sequences formed using linear combinations of multiplicative characters of finite fields are exhibited that, when compared to random sequence pairs, simultaneously achieve significantly lower mean square autocorrelation values (for each sequence in the pair) and significantly lower mean square crosscorrelation values. If we define crosscorrelation merit factor analogously to the usual merit factor for autocorrelation, and if we define demerit factor as the reciprocal of merit factor, then randomly selected binary sequence pairs are known to have an average crosscorrelation demerit factor of $1$. Our constructions provide sequence pairs with crosscorrelation demerit factor significantly less than $1$, and at the same time, the autocorrelation demerit factors of the individual sequences can also be made significantly less than $1$ (which also indicates better than average performance). The sequence pairs studied here provide combinations of autocorrelation and crosscorrelation performance that are not achievable using sequences formed from single characters, such as maximal linear recursive sequences (m-sequences) and Legendre sequences. In this study, exact asymptotic formulae are proved for the autocorrelation and crosscorrelation merit factors of sequence pairs formed using linear combinations of multiplicative characters. Data is presented that shows that the asymptotic behavior is closely approximated by sequences of modest length.Comment: 47 page