Low Autocorrelation Binary Sequences Research Articles

Evidence of scaling advantage for the quantum approximate optimization algorithm on a classically intractable problem.

The quantum approximate optimization algorithm (QAOA) is a leading candidate algorithm for solving optimization problems on quantum computers. However, the potential of QAOA to tackle classically intractable problems remains unclear. Here, we perform an extensive numerical investigation of QAOA on the low autocorrelation binary sequences (LABS) problem, which is classically intractable even for moderately sized instances. We perform noiseless simulations with up to 40 qubits and observe that the runtime of QAOA with fixed parameters scales better than branch-and-bound solvers, which are the state-of-the-art exact solvers for LABS. The combination of QAOA with quantum minimum finding gives the best empirical scaling of any algorithm for the LABS problem. We demonstrate experimental progress in executing QAOA for the LABS problem using an algorithm-specific error detection scheme on Quantinuum trapped-ion processors. Our results provide evidence for the utility of QAOA as an algorithmic component that enables quantum speedups.

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.

Solving unconstrained binary polynomial programs with limited reach: Application to low autocorrelation binary sequences

Unconstrained Binary Polynomial Programs (UBPs) are a class of optimization problems relevant in a broad array of fields. In this paper, we examine an example from communication engineering, namely low autocorrelation binary sequences and propose a new dynamic programming approach that is particularly effective on UBP instances that have a limited so-called reach, which is a metric that states the maximum difference between any two variable indices across all monomials in the UBP. Based on the reach, the dynamic programming approach decomposes the problem into a number of overlapping stages that can be solved in parallel. On a set of publicly available low autocorrelation binary sequence problems, we demonstrate the superiority of the approach by showing that the method solves to optimality for the first time several previously unsolved instances. In particular, we provide a direct comparison between the proposed method and a modern version of a previously proposed dynamic program for UBPs. We give a detailed analysis of the connection between the two different algorithms and demonstrate that the advantage of the proposed dynamic program is in its ability to implicitly identify the multilinear polynomials that are required in the recursive steps of the two dynamic programs. For perspective, a comparison to several other methods is also provided.

Computational Search of Long Skew-symmetric Binary Sequences with High Merit Factors

In this paper, we present a computational search for best-known merit factors of longer binary sequences with an odd length. Finding low autocorrelation binary sequences with optimal or suboptimal merit factors is a very difficult optimization problem. An improved version of the heuristic algorithm is presented and tackled to search for aperiodic binary sequences with good autocorrelation properties. High-performance computations with the execution of our stochastic algorithmto search skew-symmetric binary sequences with high merit factors. After experimental work, as results, we present new binary sequences with odd lengths between 201 and 303 that are skew-symmetric and have the merit factor F greater than 8.5. Moreover, an example of a binary sequence having F > 8 has been found for all odd lengths between 201 and 303. The longest binary sequence with F > 9 found to date is of length 255.

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.

Complex networks analysis of the energy landscape of the low autocorrelation binary sequences problem

We provide an up-to-date view of the structure of the energy landscape of the low autocorrelation binary sequences problem, a typical representative of the NP-hard class. To study the landscape features of interest we use the local optima network methodology through exhaustive extraction of the optima graphs for problem sizes up to 24. Several metrics are used to characterize the networks: number and type of optima, optima basins structure, degree and strength distributions, shortest paths to the global optima, and random walk-based centrality of optima. Taken together, these metrics provide a quantitative and coherent explanation for the difficulty of the low autocorrelation binary sequences problem and provide information that could be exploited by optimization heuristics for this problem, as well as for a number of other problems having a similar configuration space structure.

Solving unconstrained 0-1 polynomial programs through quadratic convex reformulation

We propose a method called Polynomial Quadratic Convex Reformulation (PQCR) to solve exactly unconstrained binary polynomial problems (UBP) through quadratic convex reformulation. First, we quadratize the problem by adding new binary variables and reformulating (UBP) into a non-convex quadratic program with linear constraints (MIQP). We then consider the solution of (MIQP) with a specially-tailored quadratic convex reformulation method. In particular, this method relies, in a pre-processing step, on the resolution of a semi-definite programming problem where the link between initial and additional variables is used. We present computational results where we compare PQCR with the solvers Baron and Scip. We evaluate PQCR on instances of the image restoration problem and the low auto-correlation binary sequence problem from MINLPLib. For this last problem, 33 instances were unsolved in MINLPLib. We solve to optimality 10 of them, and for the 23 others we significantly improve the dual bounds. We also improve the best known solutions of many instances.

Low Autocorrelation Binary Sequences: Best-Known Peak Sidelobe Level Values

Binary sequences are widely used in many practical fields, such as radar applications, telecommunications and cryptography. Finding low autocorrelation binary sequences with good peak side-lobe level (PSL) values is a difficult optimization problem. In this paper we present an improved heuristic algorithm for searching low autocorrelation PSL sequences. A heuristic algorithm can find a sequence with a PSL value, which is not necessarily optimal, but is usually near optimal, and the algorithm finds it in a reasonable amount of time. In the experimental work we applied our algorithm to find binary sequences with low PSL values, and made a comparison with the state-of-the-art algorithms from literature. With our algorithm many sequences with the currently best-known PSL values have been improved. We found new sequences with better, i.e., lower, PSL values.

A study of the loops control for reconfigurable computing with OpenCL in the LABS local search problem

In this article, we study the steepest descent local search (SDLS) algorithm that is used as the improvement step in the memetic algorithms for the search of low autocorrelation binary sequences (LABS). We address the method of reconfigurable computing, as the algorithm is of the field programmable gate array (FPGA) type as it features the integer operations, bit-wise data representation, regular execution flow, and huge computational complexity. It contains four levels of nested loops, but its direct parallel implementation as a custom processor leads to typical problems because the loops expose a dynamic range and too many iterations. This inhibits a simple parallel data path that is typically produced by the method of the loop unrolling. We have examined the four architectures that mitigate the found obstacles, and we provide the results of their implementation. The solutions take advantages of the loop pipelining, reordering of the loops, and dynamic reconfiguration. The recently available development tool was involved in our study as we have used the OpenCL (OCL) platform for FPGAs to draw practical conclusions. The given proposals are characterized by their performance and capacity for a problem size. Consequently, the speed/size trade-off is highlighted, as an FPGA size is a design constraint. The performance of the FPGA-based solutions is compared to the CPU speed, and the maximum reported speed-up is 750. Readers can further develop and/or use the presented OCL solutions for efficient LABS discovery as we provide the corresponding software repository.

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: 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*1.1504L, 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.

Generating Fluttering Patterns with Low Autocorrelation for Coded Exposure Imaging

The performance of coded exposure imaging critically depends on finding good binary sequences. Previous coded exposure imaging methods have mostly relied on random searching to find the binary codes, but that approach can easily fail to find good long sequences, due to the exponentially expanding search space. In this paper, we present two algorithms for generating the binary sequences, which are especially well suited for generating short and long binary sequences, respectively. We show that the concept of low autocorrelation binary sequences, which has been successfully exploited in the field of information theory, can be applied to generate shutter fluttering patterns. We also propose a new measure for good binary sequences. Based on the new measure, we introduce two new algorithms for coded exposure imaging - a modified Legendre sequence method and a memetic algorithm. Experiments using both synthetic and real data show that our new algorithms consistently generate better binary sequences for the coded exposure problem, yielding better deblurring and resolution enhancement results compared to previous methods of generating the binary codes.

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 .

An Algorithmic Approach towards Construction of Long Binary Sequences using Modified Jacobi Sequences

Construction of long low autocorrelation binary sequences (LABS) is a complex process which involves many limitations. LABS have many practical applications. In pulse coding schemes, sequences with low autocorrelation side lobe energies are required to reduce the noise and to increase the capability of radars to detect multiple targets. In literature, numerous techniques were employed to solve the LABS problem. For short length sequences, search algorithms can be applied as the search space is manageable. But in our case of long length binary sequences, construction methods are suitable. The major limitations of search algorithms are time and computational power. DH Green [1] in their research utilized modified Jacobi sequences to construct merit factors for long binary sequences. In our case, we used the same construction methods and applied them to various search algorithms. We obtained better results with this implementation. We achieved a merit factor of 6.4534 whereas Green [1] managed to 5.99.

Improvement of Long Binary Sequence Merit Factors using Modified Legendre Algorithms

Low autocorrelation binary sequence (LABS) detection is a classic problem in the literature. We use these sequences in many real-life applications. The detection of these sequences involves many problems. In the literature, various methods have been developed to approach the LABS issue. Based on the length of the sequence, an appropriate method can be selected and implemented. For short length sequences, linear search is possible and as the length increases we can implement various stochastic optimization algorithms. In our case that is for long binary sequences, we can use construction methods. Kristiansen and Parker [1] in their work have shown that Legendre sequences with periodic rotation can achieve a merit factor of 6.34.We have applied these Legendre sequences to steepest descent and prime step algorithms with some modifications. We call these techniques as modified Legendre algorithms. Using these improved methods we were able to achieve a merit factor of 6.4245 for long binary sequences.

New evolutionary search for long low autocorrelation binary sequences

Binary sequences with low aperiodic autocorrelation levels, defined in terms of the peak sidelobe level (PSL) and/or merit factor, have many important engineering applications, such as radars, sonars, spread spectrum communications, system identification, and cryptography. Searching for low autocorrelation binary sequences (LABS) is a notorious combinatorial problem, and has been chosen to form a benchmark test for constraint solvers. Due to its prohibitively high complexity, an exhaustive search solution is impractical, except for relatively short lengths. Many suboptimal algorithms have been introduced to extend the LABS search for lengths of up to a few hundred. In this paper, we address the challenge of discovering even longer LABS by proposing an evolutionary algorithm (EA) with a new combination of several features, borrowed from genetic algorithms, evolutionary strategies (ES), and memetic algorithms. The proposed algorithm can efficiently discover long LABS of lengths up to several thousand. Record-breaking minimum peak sidelobe results of many lengths up to 4096 have been tabulated for benchmarking purposes. In addition, our algorithm design can be easily adapted to tackle various extensions of the LABS problem, say, with a generic sidelobe criterion and/or for possibly nonbinary sequences.

An Electromagnetism-Like Approach for Solving the Low Autocorrelation Binary Sequence Problem

In this paper an electromagnetism-like approach (EM) for solving the low autocorrelation binary sequence problem (LABSP) is applied. This problem is a notoriously difficult computational problem and represents a major challenge to all search algorithms. Although EM has been applied to the topic of optimization in continuous space and a small number of studies on discrete problems, it has potential for solving this type of problems, since movement based on the attraction-repulsion mechanisms combined with the proposed scaling technique directs EM to promising search regions. Fast implementation of the local search procedure additionally improves the efficiency of the overall EM system.

Search for Low Autocorrelation Binary Sequences

Binary sequences with low autocorrelation are very important for many applications in communication and radar systems, while the construction of such sequences is one of the most challenging problems in signal design theory. In this letter we propose a Local Optimum Shift (LOS) algorithm to search for low autocorrelation binary sequences. The LOS uses an ingenious operation to exploit and explore the search space efficiently. Some new binary sequences with the best known Peak Sidelobe Level (PSL) are found out by the LOS. For binary sequences of lengths up to 100, the LOS can generate promising sequences with higher merit factor and lower PSL quickly.

A Mixed Integer Quadratic Programming Model for the Low Autocorrelation Binary Sequence Problem

In this paper the low autocorrelation binary sequence problem (LABSP) is modeled as a mixed integer quadratic programming (MIQP) problem and proof of the model’s validity is given. Since the MIQP model is semidefinite, general optimization solvers can be used, and converge in a finite number of iterations. The experimental results show that IQP solvers, based on this MIQP formulation, are capable of optimally solving general/skew-symmetric LABSP instances of up to 30/51 elements in a moderate time. ACM Computing Classification System (1998): G.1.6, I.2.8.

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=N22E, 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 N2 and N−12 respectively, making the merit factor equal to N and N2N−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).