Skew-symmetric Sequences Research Articles

Parallel self-avoiding walks for a low-autocorrelation binary sequences problem

A low-autocorrelation binary sequences problem with a high figure of merit factor represents a formidable computational challenge. An efficient parallel computing algorithm is required to reach the new best-known solutions for this problem. Therefore, we developed the sokolskew solver for the skew-symmetric search space. The developed solver takes the advantage of parallel computing on graphics processing units. The solver organized the search process as a sequence of parallel and contiguous self-avoiding walks and achieved a speedup factor of 387 compared with lssOrel, its predecessor. The sokolskew solver belongs to stochastic solvers and cannot guarantee the optimality of solutions. To mitigate this problem, we established the predictive model of stopping conditions according to the small instances for which the optimal skew-symmetric solutions are known. With its help and 99% probability, the sokolskew solver found all the known and seven new best-known skew-symmetric sequences for odd instances from L=121 to L=223. For larger instances, the solver cannot reach 99% probability within our limitations, but it still found several new best-known binary sequences. We also analyzed the trend of the best merit factor values, and it shows that as sequence size increases, the value of the merit factor also increases, and this trend is flatter for larger instances.

A Hybrid Modified Sine Cosine Algorithm Using Inverse Filtering and Clipping Methods for Low Autocorrelation Binary Sequences

The essential purpose of radar is to detect a target of interest and provide information concerning the target's location, motion, size, and other parameters. The knowledge about the pulse trains’ properties shows that a class of signals is mainly well suited to digital processing of increasing practical importance. A low autocorrelation binary sequence (LABS) is a complex combinatorial problem. The main problems of LABS are low Merit Factor (MF) and shorter length sequences. Besides, the maximum possible MF equals 12.3248 as infinity length is unable to be achieved. Therefore, this study implemented two techniques to propose a new metaheuristic algorithm based on Hybrid Modified Sine Cosine Algorithm with Cuckoo Search Algorithm (HMSCACSA) using Inverse Filtering (IF) and clipping method to achieve better results. The proposed algorithms, LABS-IF and HMSCACSA-IF, achieved better results with two large MFs equal to 12.12 and 12.6678 for lengths 231 and 237, respectively, where the optimal solutions belong to the skew-symmetric sequences. The MF outperformed up to 24.335% and 2.708% against the state-of-the-art LABS heuristic algorithm, xLastovka, and Golay, respectively. These results indicated that the proposed algorithm's simulation had quality solutions in terms of fast convergence curve with better optimal means, and standard deviation.

Optimal search for binary skew-symmetric sequences with minimal levels of side lobes

Optimal search for binary skew-symmetric sequences with minimal levels of side lobes

A Heuristic Algorithm for a Low Autocorrelation Binary Sequence Problem With Odd Length and High Merit Factor

A low autocorrelation binary sequence (LABS) problem is a hard combinatorial problem and its solutions are important in many practical applications. Till now, the largest best-known skew-symmetric sequence with merit factor greater than 9 had a length of 189. In this paper, a new heuristic algorithm is presented for the LABS problem. The proposed algorithm stores promising solutions and this mechanism enables the algorithm to perform local searches on these solutions in a systematic way. Our algorithm was tested on skew-symmetric sequences and the obtained results are compared with results of the state-of-the-art algorithms. The proposed algorithm was able to find some new best-known skew-symmetric solutions with merit factor greater than 9 in sequence lengths over 200. The obtained results improve the suggestion from 1985 (Beenker et al. ) and 1987 (Bernasconi) greatly, where the merit factor is approximately equal to 6 for long skew-symmetric sequences with length up to 199. Now, the largest best-known skew-symmetric sequence with merit factor greater than 9 has the length 225. Additionally, now all merit factors are greater than 8.5 on the interval from 159 up to 225 for odd lengths.

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 . We computed all optimal sequences for and all optimal skewsymmetric sequences for .

Autocorrelations of Binary Sequences and Run Structure

We analyze the connection between the autocorrelation of a binary sequence and its run structure given by the run length encoding. We show that both the periodic and the aperiodic autocorrelation of a binary sequence can be formulated in terms of the run structure. The run structure is given by the consecutive runs of the sequence. Let C = (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,..., C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) denote the autocorrelation vector of a binary sequence and Δ the difference operator. We prove that the th component of Δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (C) can be directly calculated by using the consecutive runs of total length k. In particular, this shows that the th autocorrelation is already determined by all consecutive runs of total length l <; k.In the aperiodic case, we show how the run vector can be efficiently calculated and give a characterization of skew-symmetric sequences in terms of their run length encoding.

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 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.

Finding low autocorrelation binary sequences with memetic algorithms

This paper deals with the construction of binary sequences with low autocorrelation, a very hard problem with many practical applications. The paper analyzes several metaheuristic approaches to tackle this kind of sequences. More specifically, the paper provides an analysis of different local search strategies, used as stand-alone techniques and embedded within memetic algorithms. One of our proposals, namely a memetic algorithm endowed with a Tabu Search local searcher, performs at the state-of-the-art, as it consistently finds optimal sequences in considerably less time than previous approaches reported in the literature. Moreover, this algorithm is also able to provide new best-known solutions for large instances of the problem. In addition, a variant of this algorithm that explores only a promising subset of the whole search space (known as skew-symmetric sequences) is also analyzed. Experimental results show that this new algorithm provides new best-known solutions for very large instances of the problem.

Two binary sequence families with large merit factor

We calculate the asymptotic merit factor, under all rotations of sequence elements, of two families of binary sequences derived from Legendre sequences. The rotation is negaperiodic for the first family, and periodic for the second family. In both cases the maximum asymptotic merit factor is 6. As a consequence, we obtain the first two families of skew-symmetric sequences with known asymptotic merit factor, which is also 6 in both cases.

Low autocorrelation binary sequences: exact enumeration and optimization by evolutionary strategies

We investigate skew-symmetric sequences with chain lengths up to N = 71, giving a complete table of all merit factors F≥7 and their associated configurations. We also calculate the exact thermodynami-cal properties of shorter chains (N≤55). We then introduce an evolutionary strategy, describing the properties of our search algorithm and comparing our results to those of other heuristic methods such as simulated annealing. We find the highest merit factors ever reached for chains of length 81≤N≤201.

A new search for skewsymmetric binary sequences with optimal merit factors

Previous results on skewsymmetric sequences with low autocorrelation obtained by making exhaustive searches for sequences with optimal merit factor are extended up to length 69. A sieve that required both the symmetric and antisymmetric components of a skewsymmetric sequence to have high merit factor to make limited searches up to length 117 was used, and many sequences with merit factor greater than 9 were found. >

Sieves for low autocorrelation binary sequences

The value 2e^{2} is calculated by approximations as a conjectured asymptotic limit for the best central to sidelobe energy ratio of very long low autocorrelation binary sequences. The same bound is calculated by approximation for skewsymmetric sequences, hence it is concluded that the requirement of skewsymmetry constitutes a powerful and effective sieve for the search for near-optimal sequences. A second sieve based on the use of complementary sequences is discussed and shown to be quite effective for sequence lengths up to 31, and its investigation for longer lengths by means of computers is suggested. A third and a fourth sieve, based on the selection of restricted classes of complementary sequences, are also discussed.