Golomb Ruler Research Articles

Solomon W. Golomb—Mathematician, Engineer, and Pioneer

In this paper, we present some fundamental concepts and theoretical advances attributable to Solomon Golomb, together with the history and applications of this paper to communications, coding, and cryptography, along with some long-standing conjectures. Examples include the first engineering problem relating to feedback shift-register sequences that Sol Golomb was asked to solve in the mid-1950s. This paper covers $m$ -sequences and Golomb’s three randomness postulates, the cross-correlation of $m$ -sequences, the exp-Golomb code, the Golomb ruler, Costas arrays, Golomb invariants, polyominoes, the distribution of prime numbers, and irreducible polynomials.

A Career in Engineering

During the summers (1951–1954) that I was a graduate student in “pure mathematics,” I worked in the Systems Engineering Section of the Glenn L. Martin Company. I began to notice the applicability of supposedly pure topics like prime number theory and finite field theory to problems in communications. My first major applied effort involved developing the theory of “Shift Register Sequences.” (My book with this title will soon see its third edition, with a third publisher.) Much of my work has been in response to practical questions I was asked, for which I had the necessary mathematical tools. These topics have included comma-free codes, Costas arrays, Tuscan squares, Golomb rulers, zero-sidelobe radar, etc. My shift register work has had the broadest impact: to cell phone signals, the GPS system, error-correcting codes, radar, cryptography, etc. I am fortunate to have lived long enough to get some recognition for my work.

Design and performance analysis of modified two dimensional Golomb code for optical code division multiple access networks

In this work, a two dimensional (2D) wavelength/time code is developed. The 64 bit 2D code is constructed by a technique based on folding of Golomb rulers referred in this work as modified Golomb code (2D MGC) and the proposed 2D code is generated using Java 8.1 software. The performance of the proposed 2D MGC is evaluated in terms of bit error rate (BER), received signal power, and time domain analysis of multiple users. In this experiment, it is found that the BER increases with multiple numbers of simultaneous users due to multiple access interference. The proposed 2D code yields better network performance in terms of BER compared to that of different existing 2D OCDMA codes. In this work, mathematical analysis is also carried out to prove that the proposed 64 bit code is optimal satisfying auto and cross correlation properties. Furthermore, the complexity analysis of the proposed 64 bit optical code division multiple access (OCDMA) code is analysed. The OCDMA model is validated using the optisystem software.

Constraint-based search for optimal Golomb rulers

Finding optimal Golomb rulers is an extremely challenging combinatorial problem. The distance between each pair of mark is unique in a Golomb ruler. For a given number of marks, an optimal Golomb ruler has the minimum length. Golomb rulers are used in application areas such as X-ray crystallography, radio astronomy, information theory, and pulse phase modulation. The most recent optimal Golomb ruler search algorithm hybridises a range of techniques such as greedy randomised adaptive search, scatter search, tabu search, clustering techniques, and constraint programming, and obtains optimal Golomb rulers of up to 16 marks with very low success rates. In this paper, we provide tight upper bounds for Golomb ruler marks and present heuristic-based effective domain reduction techniques. Using these along with tabu and configuration checking meta-heuristics, we then develop a constraint-based multi-point local search algorithm to perform a satisfaction search for optimal Golomb rulers of specified length. We then present an algorithm to perform an optimisation search that minimises the length using the satisfaction search repeatedly. Our satisfaction search finds optimal Golomb rulers of up to 19 marks while the optimisation search finds up to 17 marks.

Nature–inspired metaheuristic algorithms to find near–OGR sequences for WDM channel allocation and their performance comparison

Abstract Nowadays, nature–inspired metaheuristic algorithms are most powerful optimizing algorithms for solving the NP–complete problems. This paper proposes three approaches to find near–optimal Golomb ruler sequences based on nature–inspired algorithms in a reasonable time. The optimal Golomb ruler (OGR) sequences found their application in channel–allocation method that allows suppression of the crosstalk due to four–wave mixing in optical wavelength division multiplexing systems. The simulation results conclude that the proposed nature–inspired metaheuristic optimization algorithms are superior to the existing conventional and nature–inspired algorithms to find near–OGRs in terms of ruler length, total optical channel bandwidth, computation time, and computational complexity. Based on the simulation results, the performance of proposed different nature–inspired metaheuristic algorithms are being compared by using statistical tests. The statistical test results conclude the superiority of the proposed nature–inspired optimization algorithms.

Sparse distributed sensor placement via statistical restricted isometry property

Perfect sensor coverage of large ocean volumes is an intractable problem for small N systems. Instead, this paper presents optimal placement of relatively few sensors in order to achieve coherent array processing. The sensor placement will simulate acoustic beam steering despite being a severely underdetermined problem. Utilizing the statistical restricted isometry property (StRIP), with high probability, a stable (invertable) sparse array will be formed. Additionally, sensor placement will be shown with spatial constraints such as improved field of view (resolution and side lobe suppression) in a direction of a priori interest. A comparison of the StRIP optimized network will be made with co-prime samplers, random arrays, and Wichmann and Golomb rulers. [This work was supported by ONR.]

Toward hybrid platform for evolutionary computations of hard discrete problems

Memetic agent-based paradigm, which combines evolutionary computation and local search techniques in one of promising meta-heuristics for solving large and hard discrete problem such as Low Autocorrellation Binary Sequence (LABS) or optimal Golomb-ruler (OGR). In the paper as a follow-up of the previous research, a short concept of hybrid agent-based evolutionary systems platform, which spreads computations among CPU and GPU, is shortly introduced. The main part of the paper presents an efficient parallel GPU implementation of LABS local optimization strategy. As a means for comparison, speed-up between GPU implementation and CPU sequential and parallel versions are shown. This constitutes a promising step toward building hybrid platform that combines evolutionary meta-heuristics with highly efficient local optimization of chosen discrete problems.

NASA Computational Case Study: Golomb Rulers and Their Applications

In this case study, the author describes Golomb rulers, their properties, construction algorithms, and applications in astrophysics and Earth sciences.

Leveraging heterogeneous parallel platform in solving hard discrete optimization problems with metaheuristics

The research reported in the paper deals with difficult black-box problems solved by means of popular metaheuristic algorithms implemented on up-to-date parallel, multi-core, and many-core platforms. In consecutive publications we are trying to show how particular population-based techniques may further benefit from employing dedicated hardware like GPGPU or FPGA for delegating different parts of the computing in order to speed it up. The main contribution of this paper is an experimental study focused on profiling of different possibilities of implementation of Scatter Search algorithm, especially delegating some of its selected components to GPGPU. As a result, a concise know-how related to the implementation of a population-based metaheuristic similar to Scatter Search is presented using a difficult discrete optimization problem; namely, Golomb Ruler, as a benchmark.

Optimal Golomb Ruler Sequences Generation for Optical WDM Systems: A Novel Parallel Hybrid Multi-objective Bat Algorithm

In real-life, multi-objective engineering design problems are very tough and time consuming optimization problems due to their high degree of nonlinearities, complexities and inhomogeneity. Nature-inspired based multi-objective optimization algorithms are now becoming popular for solving multi-objective engineering design problems. This paper proposes original multi-objective Bat algorithm (MOBA) and its extended form, namely, novel parallel hybrid multi-objective Bat algorithm (PHMOBA) to generate shortest length Golomb ruler called optimal Golomb ruler (OGR) sequences at a reasonable computation time. The OGRs found their application in optical wavelength division multiplexing (WDM) systems as channel-allocation algorithm to reduce the four-wave mixing (FWM) crosstalk. The performances of both the proposed algorithms to generate OGRs as optical WDM channel-allocation is compared with other existing classical computing and nature-inspired algorithms, including extended quadratic congruence (EQC), search algorithm (SA), genetic algorithms (GAs), biogeography based optimization (BBO) and big bang-big crunch (BB-BC) optimization algorithms. Simulations conclude that the proposed parallel hybrid multi-objective Bat algorithm works efficiently as compared to original multi-objective Bat algorithm and other existing algorithms to generate OGRs for optical WDM systems. The algorithm PHMOBA to generate OGRs, has higher convergence and success rate than original MOBA. The efficiency improvement of proposed PHMOBA to generate OGRs up to 20-marks, in terms of ruler length and total optical channel bandwidth (TBW) is 100 %, whereas for original MOBA is 85 %. Finally the implications for further research are also discussed.

Optimal Golomb Ruler Sequences asWDM Channel-Allocation Algorithm Generation: Cuckoo Search Algorithm with Mutation

With the hailing winds of development, humans are still can’t fully act individually, the dependency over the nature is indispensable. In this paper, a hybrid nature inspired based algorithm named Cuckoo search algorithm with mutation (CSAM) has been used to solve the channel–allocation problem presents in optical wavelength division multiplexing (WDM) systems. The channels can be allocated by using the concept of shortest length rulers called optimal Golomb ruler (OGR) sequences to suppress four-wave mixing (FWM) crosstalk. The simulation results reveals that computational time taken by CSAM to generate channel–allocation algorithm has been abated substantially unlike other existing nature inspired based algorithms such as Genetic algorithms (GAs), Biographically based optimization (BBO), and Cuckoo search algorithm (CSA). The simulation results obtained by proposed hybrid algorithm demonstrates better and efficient in terms of length of the ruler, total channel bandwidth, and bandwidth expansion factor compared to simple classical approaches such as Extended quadratic congruence (EQC) and Search algorithm (SA) and other nature inspired based algorithms.

Optimization Algorithms for GeneratingGolomb Ruler Sequences A Comparison

An undesirable nonlinear effect which degrades system performance significantly is acknowledged as Four Wave Mixing (FWM). In the optical communication systems, the FWM crosstalk is reduced by using unequally spaced channel allocation method. This channel allotment method is constructed on optimal Golomb ruler (OGR) which not only maintains the bandwidth efficiency but also decreases the FWM effect. To generate and optimize Golomb ruler sequence, classic approaches as well as nature- inspired algorithms have been used by various researchers. In this paper, some of the algorithms are reviewed and compared.

Superresolution Multipoint Ranging With Optimized Sampling via Orthogonally Designed Golomb Rulers

We consider the problem of performing ranging measurements between a source and multiple receivers efficiently and accurately, as required by distance-based wireless localization systems. To this end, a new multipoint ranging algorithm is proposed, which is obtained by adapting superresolution techniques to the ranging problem, using for the sake of illustration the specific cases of time of arrival (ToA) and phase-difference of arrival (PDoA), unified under the same mathematical framework. The resulting nonparametric algorithm handles multipoint ranging in an efficient manner by employing an orthogonalized nonuniform sampling scheme optimized via Golomb rulers. Since the approach requires the design of mutually orthogonal sets of Golomb rulers with equivalent properties—a problem that founds no solution in current literature—a new genetic algorithm to accomplish this task is presented, which is also found to outperform the best known alternative when used to generate a single ruler. Finally, a Cramer-Rao lower bound (CRLB) analysis of the overall optimized multipoint ranging solution is performed, which together with a comparison against simulation results validates the proposed techniques.

Construction of High-Performance Array-Based Non-Binary LDPC Codes With Moderate Rates

Based on arrays, excellent low-density parity-check (LDPC) codes with high rates have been constructed. In this letter, we propose to construct high-performance nonbinary LDPC (NBLDPC) codes with moderate rates based on arrays. The proposed construction combines masking and column selection. First, we construct a masking matrix $\mathbf{M}$ based on asymptotic thresholds and certain error-prone substructures. Second, we generate a number of column selection sets (CSS) based on Golomb rulers. Based on $\mathbf{M}$ , we construct a parity-check matrix for each CSS. Third, we compute the cycle distribution of each parity-check matrix and select the one with the best cycle distribution as output. To assess the effectiveness of the proposed construction, error performances of several NBLDPC codes are presented. Comparisons with other codes are also carried out to show the advantages of our construction.

Simulation and Analysis of Optimum Golomb Ruler Based 2D Codes for OCDMA System

The need for high speed communications networks has led the research communities and industry to develop reliable, scalable transatlantic and transpacific fiber-optic communication links. In this paper the optimum Golomb ruler based 2D OCDMA codes has been demonstrated. An OCDMA system based on the discussed 2D codes is designed and simulated on Optisystem. The encoder and decoder structure of OCDMA system have been designed using filter and time delays. Further the performance is analysed for various parameter such as bit rate, number of users, BER (Bit Error Rate), quality factor, eye diagram and signal diagram. The system is analyzed for up to 18 users at 1 Gbps and 1.25 Gbps bit rate.

A Novel Hybrid Multi–objective BB–BC based Channel Allocation Algorithm to Reduce FWM Crosstalk and its Comparative Study

is a good source of inspirations for us. The algorithms developed from the nature are most powerful algorithms for optimizing many complex engineering design problems having multiple objectives (multi-objective). This paper presents an hybrid algorithm based on Multi-objective Big bang-Big Crunch (MOBB-BC) nature-inspired optimization algorithm with Genetic crossover and Differential evolution (DE) mutation operators for solving the minimum length ruler called Optimal Golomb ruler (OGR) as channel-allocation problem to reduce four-wave mixing crosstalk (FWM) effects in optical wavelength division multiplexing (WDM) systems. The comparative study of simulation results obtained by proposed hybrid Multi-objective BB-BC (HMOBB-BC) algorithm demonstrates better and efficient generation of OGRs in a reasonable computational time compared to simple BB-BC algorithm and one of the existing nature-inspired algorithms i.e. Genetic algorithm (GA). Also, the proposed hybrid algorithm outperforms the two existing conventional algorithms i.e. Extended quadratic congruence (EQC) and Search algorithm (SA), in terms of ruler length and total channel bandwidth.

Inglés

A set of positive integers A is called a g-Golomb ruler if the difference between two distinct elements of A is repeated at most g times. This definition is a generalization of the Golomb ruler (g = 1). In this paper we construct g-Golomb ruler from Golomb ruler and we prove two theorems about extremal functions associated with this sets. Keywords: Sidon sets, B2 sets, Golomb ruler. To cite this article: Y. Caicedo, C.A. Martos, C.A. Trujillo, g-Golomb, Rev. Integr. Temas Mat. 33 (2015), No. 2, 161–172.

Optimal signals of Golomb ruler class for spectral measurements at EKB SuperDARN radar: Theory and experiment

The requirements for an optimal sequence for SuperDARN radar are stated. Obtained quasioptimal signals up to 26-pulse sequence are based on optimal and nearly optimal Golomb rulers. It is shown that standard 7-pulse SuperDARN sequence is one of the quasioptimal sounding signals, but 8-pulse kascan SuperDARN signal is not. The characteristics are calculated for quasioptimal sounding sequences up to 26-pulse one. It is shown that the most effective signals for the spectral measurements are 10, 12, 14, 18, 20, 24 and 26 pulse sequences. We present the results of the first spectral measurements at the EKB SuperDARN radar with quasioptimal 8- and 9-pulse sequences, as well as with the optimal 10-and 12-pulse sequences. A comparison with the results obtained with standard 7-pulse SuperDARN sequence were made. The continuity of results in amplitude and velocity is shown, the expected improvement in the spectral width are also demonstrated.

GPGPU for Difficult Black-box Problems

Difficult black-box problems arise in many scientific and industrial areas. In this paper, efficient use of a hardware accelerator to implement dedicated solvers for such problems is discussed and studied based on an example of Golomb Ruler problem. The actual solution of the problem is shown based on evolutionary and memetic algorithms accelerated on GPGPU. The presented results prove that GPGPU outperforms CPU in some memetic algorithms which can be used as a part of hybrid algorithm of finding near optimal solutions of Golomb Ruler problem. The presented research is a part of building heterogenous parallel algorithm for difficult black-box Golomb Ruler problem.

Investigating wavelength allocation and suitable modulation format for 2-d wavelength/time and 3-d wavelength/time/space codes in order to enhance system performance

In this paper the design, implementation and performance analysis of 2-dimensional and 3-dimensional wavelength/time/space codes in OCDMA system have been presented using folding of Golomb rulers. It is shown that BER increases with increase in number of active users when number of decoder’s increases on receiver side. Further, the optical encoding/decoding for 90 km length of fiber without amplification has been reported with good quality factor (Q) with 32 users while maintaining bit-error rate (BER) < 1e-11 for the correctly decoded signal. The comparison of 3-D single pulse differential detection (SPDD) with designed 2-D wavelength-time (W/T) matrix codes shows that the 3-D SPDD codes perform significantly better.