Golomb Ruler Research Articles

On the size of finite Sidon sets

UDC 519.1 A Sidon set (also called a Golomb ruler) is a B 2 sequence and a 1 -thin set is a set of integers containing no nontrivial solutions to the equation a + b = c + d . We improve the lower bound for the diameter of a Sidon set with k elements, namely, if k is sufficiently large and 𝒜 is a Sidon set with k elements, then ⅆ i a m ( 𝒜 ) ≥ k 2 - 1.99405 k 3 / 2 . Alternatively, if n is sufficiently large, then the cardinality of the largest subset of { 1,2 , … , n } , which is a Sidon set, does not exceed n 1 / 2 + 0.99703 n 1 / 4 .

Isotope interface engineering for thermal transport suppression in cryogenic graphene

The development of emerging technologies, such as quantum computing and semiconductor electronics, emphasizes the growing significance of thermal management at cryogenic temperatures. Herein, by designing isotope interfaces based on the Golomb ruler, we achieved effective suppression of the phonon thermal transport of cryogenic graphene. The pronounced disordering of the Golomb ruler sequence results in the stronger suppression of thermal transport compared to other sequences with the same isotope doping ratio. We demonstrated that the Golomb ruler-based isotope interfaces have strong scattering and confinement effects on phonon transport via extensive molecular dynamics simulations combined with wave packet analysis, with a proper correction for the missing quantum statistics. This work provides a new stream for the design of thermal transport suppression under cryogenic conditions and is expected to expand to other fields.

Partial reformulation-linearization based optimization models for the Golomb ruler problem

In this paper, we provide a straightforward proof of a conjecture proposed in \cite{Duxbury2021} regarding the optimal solutions of a non-convex mathematical programming model of the Golomb ruler problem. Subsequently, we investigate the computational efficiency of four new binary mixed-integer linear programming models to compute optimal Golomb rulers. These models are derived from a well-known nonlinear integer model proposed in \cite{Kocuk2019}, utilizing the reformulation-linearization technique. \modif{Finally, we provide the correct outputs of the greedy heuristic proposed in \cite{Duxbury2021} and correct some false conclusions stated or implied therein.}

Rate-compatible LDPC Codes based on Primitive Polynomials and Golomb Rulers

Rate-compatible LDPC Codes based on Primitive Polynomials and Golomb Rulers

Designs of Electronic Devices using Combinatorial Optimization

This paper involves design techniques of electronic devices by combinatorial optimization for improving the quality indices of electronic devices or systems concerning performance reliability, transformation speed, positioning precision, and functionality, using proposals based on remarkable properties and structural perfection of combinatorial configurations, such as difference sets and “Golomb rulers”. These design techniques make it possible to configure electronic devices or systems with fewer elements than at present while maintaining or improving on functionality and the other significant operating characteristics of the devices.

Suppressed thermal transport in mathematically inspired 2D heterosystems

In two-dimensional (2D) heterosystems, the contribution of coherent phonon transport makes superlattices with high interface density exhibit unexpected high thermal conductivity. Herein, inspired by mathematics, we demonstrate efficient suppression of phonon thermal transport in a 2D heterosystem constituted of graphene and hexagonal boron nitride based on the Golomb ruler sequences. This design realizes extreme suppression of thermal conductivity with a minimal number of interfaces, and without any defects or dopants. Extensive numerical simulations combined with wave packet analysis show that the Golomb ruler sequence largely cancels the coherent phonon transport. This work, as the first attempt for realizing novel thermal physics using the mathematically inspired Golomb ruler design, provides a new and efficient solution for the suppression of thermal transport in 2D heterosystems.

A construction for girth‐8 QC‐LDPC codes using Golomb rulers

In this paper, an algebraic construction of regular QC-LDPC codes by using the modular multiplication table mod P and Golomb rulers are proposed. It is proved that the proposed QC-LDPC codes based on a Golomb ruler of length L have girth at least 8 if P > 2 L $P>2L$ . The error performance of the proposed QC-LDPC codes are simulated with various Golomb rulers. The proposed codes of length around 300 from the optimal 6-mark Golomb ruler have an additional coding gain of at least 0.1 dB over 5G NR LDPC codes, 0.5 dB over those given earlier by others, both at FER 10−3. Some non-trivial techniques to increase the length of a given Golomb ruler with and without an additional mark for improving the performance of the codes from Golomb rulers up to 0.7 dB are also found.

Hadamard Aperiodic Interval Codes for Parallel-Transmission 2D and 3D Synthetic Aperture Ultrasound Imaging

We present a new set of near orthogonal codes which we call Hadamard Aperiodic Interval (HAPI) codes and demonstrate their utility for parallel multi-transmitter synthetic aperture imaging. The codes are tri-state and sparse. Locations of non-zero bits are based on marks in a sequence of aperiodic intervals, also known as a Golomb ruler. The values of non-zero bits are selected from Hadamard sequences that are mutually orthogonal. This ensures that cross-correlation sidelobe magnitudes between differing codes are bounded by unity while the autocorrelation approaches a delta function with mainlobe-to-sidelobe levels scaling with the number of non-zero bits. We use simulations to demonstrate the potential of the codes for synthetic aperture imaging. A multiplicity of transmitter elements is used to transmit codes simultaneously, with a different code for each element. Echo signals are received from a multiplicity of transducer elements in parallel. Channel data from each receiver element are cross-correlated with respective HAPI codes to estimate the transmit–receive signature associated with each transmitter–receiver pair while minimizing crosstalk. This estimate of the full transmit–receive synthetic aperture dataset is then used to form high-quality images demonstrating image quality and signal-to-noise ratio improvements over multiple flash angle imaging and synthetic aperture imaging methods for linear arrays. We also demonstrate simulated full volume synthetic aperture imaging with random sparse arrays, possible with one extended HAPI code-set transmit event.

Almost Difference Sets From Singer Type Golomb Rulers

Let <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> be an additive group of order <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula>. A <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-element subset <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is called a <inline-formula> <tex-math notation="LaTeX">$(v, k, \lambda, t)$ </tex-math></inline-formula>-almost difference set if the expressions <inline-formula> <tex-math notation="LaTeX">$g-h$ </tex-math></inline-formula>, for <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$h$ </tex-math></inline-formula> in <inline-formula> <tex-math notation="LaTeX">$D$ </tex-math></inline-formula>, represent <inline-formula> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> of the non-identity elements in <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> exactly <inline-formula> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> times and every other non-identity element <inline-formula> <tex-math notation="LaTeX">$\lambda + 1$ </tex-math></inline-formula> times. Almost difference sets are highly sought after as they can be used to produce functions with optimal nonlinearity, cyclic codes, and sequences with three-level autocorrelation. A set of positive integers <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> is called a Golomb ruler if the difference between two distinct elements of <inline-formula> <tex-math notation="LaTeX">$A$ </tex-math></inline-formula> are different. In this paper, we use Singer type Golomb rulers to construct new families of almost difference sets. Additionally, we constructed 2-adesigns from these almost difference sets.

Antenna arrays based on Golomb rulers made of copper and superconducting elements

Antenna arrays based on Golomb rulers made of copper and superconducting elements

A conjecture on a continuous optimization model for the Golomb Ruler Problem

A Golomb Ruler (GR) is a set of integer marks along an imaginary ruler such that all the distances of the marks are different. Computing a GR of minimum length is associated to many applications (from astronomy to information theory). Although not yet demonstrated to be NP-hard, the problem is computationally very challenging. This brief note proposes a new continuous optimization model for the problem and, based on a given theoretical result and some computational experiments, we conjecture that an optimal solution of this model is also a solution to an associated GR of minimum length.

Nature-Inspired Hybrid Multi-objective Optimization Algorithms in Search of Near-OGRs to Eliminate FWM Noise Signals in Optical WDM Systems and their Performance Comparison

Nowadays, the multi-objective optimizing algorithms which are inspired by nature are widely used in solving the many objectives of NP-complete engineering and industrial design problems. To eliminate one of the prevailing nonlinear optical noise mechanisms, i.e., four-wave mixing noise signals in the optical wavelength division multiplexing (WDM) systems, several unequally spaced channel-allocation algorithms were found that require increased bandwidth than the equally spaced channel-allocation. One of the bandwidths efficient, a USCA algorithm can be designed by considering the optimal Golomb rulers (OGRs) in optical WDM systems. Two nature-inspired multi-objective optimization algorithms (MOAs), namely multi-objective big bang–big crunch (MOBB-BC) optimization algorithm and multi-objective firefly optimization algorithm (MOFA), to solve NP-complete OGR problem in an optical WDM system are proposed here. Additionally, the algorithms MOBB-BC and MOFA are hybridized by introducing differential evolution (DE) based mutation and random walk features in their standard forms. These two features have the potential to explore new search space for MOAs to search for OGRs in a reasonable time. The algorithms presented here solve the two objectives in optical WDM systems, one is the occupied length by the ruler, and the other is total channel bandwidth obtained by OGRs. The obtained results suggest that the considered nature-inspired MOAs are computationally better than the other algorithms to search near-OGRs. To assert the improvement in the MOAs further, the non-parametric Friedman statistical analysis test is employed. The results conclude that for large mark values, the hybridization of MOFA with both DE mutation and Levy-flight strategies potentially performs better than the MOBB-BC and its hybrid forms to search OGR sequences. The hybrid algorithm MOBB-BC can explore OGRs up to 8-marks and near-OGRs for 9 to 20-marks with a maximum computation time of 27 h, whereas hybrid MOFA can examine up to 16-mark OGRs, but finds 17 to 20-mark near-OGRs with the maximum computation time of 20 h. The success rate of the proposed hybrid MOFA to search best OGRs up to 20-marks is 80%, whereas for hybrid algorithm MOBB-BC is 40%.

On resolvable Golomb rulers, symmetric configurations and progressive dinner parties

We define a new type of Golomb ruler, which we term a resolvable Golomb ruler. These are Golomb rulers that satisfy an additional “resolvability” condition that allows them to generate resolvable symmetric configurations. The resulting configurations give rise to progressive dinner parties. In this paper, we investigate existence results for resolvable Golomb rulers and their application to the construction of resolvable symmetric configurations and progressive dinner parties. In particular, we determine the existence or nonexistence of all possible resolvable symmetric configurations and progressive dinner parties having block size at most 13, with nine possible exceptions. For arbitrary block size k, we prove that these designs exist if the number of points is divisible by k and at least \(k^3\).

Near-Optimal g-Golomb Rulers

A set of positive integers A is called a $g$ -Golomb ruler if the difference between two distinct elements of A is repeated up to $g$ times. This definition is a generalization of the Golomb ruler $(g = 1)$ . In this paper, we obtain new constructions for $g$ -Golomb rulers from Golomb rulers, using these constructions we find some suboptimal 2 and 3-Golomb rulers with up to 124 marks and we prove two theorems related to extremal functions associated with this sets improving already known results.

Bh Sets as a Generalization of Golomb Rulers

A set of positive integers $A$ is called a Golomb ruler if the difference between two distinct elements of $A$ are different, equivalently if the sums of two elements are different ( $B_{2}$ set, Sidon set). An extension of this concept is to consider that the sum of $h $ elements in $A $ are all different, except for permutation of the summands, with $h \geq 2 $ , in this case it is said that $A $ is a set $B_{h} $ , the length of $A $ is given by $\ell (A)\,\,= \max A- \min A $ . One problem associated with this type of set is that of the optimal dense $B_{h} $ sets, that is, determining the greatest cardinal of a set $B_{h} $ contained in the integer interval $\left [{1, N }\right] $ , for this defines the function $F_{h} (N)~$ . Another problem that can be associated is the optimally short $B_{h} $ sets, that is, finding a shorter $B_{h} $ set with $m $ elements, for which the $G_{h} (m)~$ function is defined. In this paper we are going to prove that these two problems are inverse, that is, that the functions $G_{h} (m)~$ and $F_{h} (N)~$ have inverse relationships. Furthermore, the asymptotic behavior of the $G_{h} (m)~$ function is studied, obtaining some upper and lower bounds, we also obtain tables of $B_{3}$ and $B_{4}$ near-optimal up to $m = 31$ .

New results on modular Golomb rulers, optical orthogonal codes and related structures

We prove new existence and nonexistence results for modular Golomb rulers in this paper. We completely determine which modular Golomb rulers of order k exist, for all k ≤ 11 , and we present a general existence result that holds for all k ≥ 3 . We also derive new nonexistence results for infinite classes of modular Golomb rulers and related structures such as difference packings, optical orthogonal codes, cyclic Steiner systems and relative difference families.

Wideband Beam Steering Concept for Terahertz Time-Domain Spectroscopy: Theoretical Considerations

Photonic true time delay beam steering on the transmitter side of terahertz time-domain spectroscopy (THz TDS) systems requires many wideband variable optical delay elements and an array of coherently driven emitters operating over a huge bandwidth. We propose driving the THz TDS system with a monolithic mode-locked laser diode (MLLD). This allows us to use integrated optical ring resonators (ORRs) whose periodic group delay spectra are aligned with the spectrum of the MLLD as variable optical delay elements. We show by simulation that a tuning range equal to one round-trip time of the MLLD is sufficient for beam steering to any elevation angle and that the loss introduced by the ORR is less than 0.1 dB. We find that the free spectral ranges (FSRs) of the ORR and the MLLD need to be matched to 0.01% so that the pulse is not significantly broadened by third-order dispersion. Furthermore, the MLLD needs to be frequency-stabilized to about 100 MHz to prevent significant phase errors in the terahertz signal. We compare different element distributions for the array and show that a distribution according to a Golomb ruler offers both reasonable directivity and no grating lobes from 50 GHz to 1 THz.

Performance comparison of five metaheuristic nature-inspired algorithms to find near-OGRs for WDM systems

The metaheuristic approaches inspired by the nature are becoming powerful optimizing algorithms for solving NP-complete problems. This paper presents five nature-inspired metaheuristic optimization algorithms to find near-optimal Golomb ruler (OGR) sequences in a reasonable time. In order to improve the search space and further improve the convergence speed and optimization precision of the metaheuristic algorithms, the improved algorithms based on mutation strategy and Levy-flight search distribution are proposed. These two strategies help the metaheuristic algorithms to jump out of the local optimum, improve the global search ability so as to maintain the good population diversity. The OGRs found their potential application in channel-allocation method to suppress the four-wave mixing crosstalk in optical wavelength division multiplexing systems. The results conclude that the proposed algorithms are superior to the existing conventional computing algorithms i.e. extended quadratic congruence and search algorithm and nature-inspired optimization algorithms i.e. genetic algorithms, biogeography based optimization and simple big bang–big crunch to find near-OGRs in terms of ruler length, total optical channel bandwidth and computation time. The idea of computational complexity for the proposed algorithms is represented through the Big O notation. In order to validate the proposed algorithms, the non-parametric statistical Wilcoxon analysis is being considered.

Synthesis of Symmetric Sounding Sequences for Ekaterinburg Coherent Decameter Radar

Optimal sounding sequences described by a pair of mutually symmetric Golomb rulers lasting from 12 to 18 pulses were synthesized within the generalization of the previously developed 13-pulse sounding sequence of SuperDARN radars. These sequences are pairs of identical sequences that are mutually symmetric in time, are separated by a lag, and do not contain an additional pulse used earlier to measure the power profile of the scattered signal. The optimal subsequences are sought by exhaustive search over the class of optimal and nearly optimal Golomb rulers. The optimality criterion is provision of the maximum information content of the received signal in the sense of a high spectral resolution with minimum accumulation time, as well as the requirement of its efficiency for measuring the power profile of the scattered signal. The work demonstrated the potential efficiency of one of the constructed signals (16-element one) for spectral measurements using the Ekaterinburg coherent decameter radar.

A Comparative Study of Nature-Inspired Metaheuristic Algorithms in Search of Near-to-optimal Golomb Rulers for the FWM Crosstalk Elimination in WDM Systems

ABSTRACTNowadays, nature-inspired metaheuristic algorithms are the most powerful optimizing algorithms for solving NP-complete problems. This paper proposes five recent approaches to find near-optimal Golomb ruler (OGR) sequences based on nature-inspired algorithms in a reasonable time. The optimal Golomb ruler sequences found their application in channel-allocation method that allows suppression of the crosstalk due to four-wave mixing (FWM) in optical wavelength division multiplexing (WDM) systems. The simulation results conclude that the proposed nature-inspired metaheuristic optimization algorithms are superior to the existing conventional computing algorithms, i.e., Extended Quadratic Congruence (EQC) and Search algorithm (SA) and nature-inspired algorithms, i.e., Genetic algorithms (GAs), Biogeography-based optimization (BBO) and simple Big bang–Big crunch (BB-BC) optimization algorithm to find near-OGRs in terms of ruler length, total optical channel bandwidth and computation time.