Gray Code

The quality of manufactured products usually needs to be veri®ed. This paper presents an advanced CAQ approach to compare manufactured objects with reference data from underlying CAD models. First, an overview about the current state-of-the-art in optical 3D measurement techniques is given. After that the research method adopted in this paper is discussed. Furthermore, a software prototype of the presented approach in which a stripe projection system with combined gray-code and phase shifting is described. With this equipment, 3D shapes of objects or manufactured products can be measured. In order to compare the 3D data (represented in sensor coordinate system) a registration to the CAD coordinate system is needed. At ®rst, the selection of a starting point for the orientation parameters is described. For the registration process different numerical algorithms are used to minimize a distance function. To achieve a better performance, an optimization process based on 3D voxel arrays is introduced. After the registration process, several parameters for the kind of geometric displacement can be calculated and visualized. For objects that cannot be measured from one direction, a pair-wise registration as well as a global registration have been developed. Furthermore, some rapid prototyping examples to which our CAQ approach has been applied are presented. Those examples show that our method works well in practice. Finally, some application ®elds for the CAQ approach presented here are outlined.

This paper presents two polynomial-complexity techniques for assigning Gray-like binary labels to arbitrary Grassmannian constellations. In the first technique, the constellation of interest, 𝒞, is matched directly to an auxiliary constellation that can be readily Gray labelled. The optimal matching in this technique can be obtained efficiently, but its application is limited to cases in which an auxiliary constellation with a geometric structure that resembles that of 𝒞 can be identified. In the second technique no auxiliary constellation is required and the labels are generated by matching the distance spectrum of 𝒞 with that of a hypothetical constellation that is assumed to be perfectly Gray labelled. Optimal matching in this case is computationally prohibitive. Instead, an efficient suboptimal matching algorithm is proposed. When compared with several existing schemes, the proposed labellings provide better performance in both uncoded and BICM-based non-coherent MIMO systems with iterative demapping and decoding (IDD). Furthermore, with the proposed labels, the Grassmannian-based BICM-IDD scheme performs better than a training-based counterpart that employs the Golden code and optimal demapping.

In this study, images of cyclic codes in two variable rings with coefficient field p F are detected. A special ring in two variables is defined under certain conditions. The Gray images of the cyclic codes over this ring are investigated. Relations between the codes over this ring and the codes over a finite chain ring in one variable are obtained via a Gray map. Another Gray map from the finite chain ring to a finite field is defined and then the images of cyclic codes are obtained. It is obtained that the Gray image of a cyclic code over R with length n .

This paper deals with constacyclic codes over a new ring. Gray images of these codes over those rings are obtained. The relations between these codes and their Gray images are given. Cyclic and constacyclic codes over finite rings with two variables were investigated in many different studies. Transformations preserving distances from these rings to another finite ring or finite fields are studied. Construction of (1 ) v  -contacyclic codes over the ring was studied in [6] and Gray images of these codes are given. Cyclic codes, quasi-cyclic codes over 2 2 2 2 u v uv        and their dual were studied by different authors [3,6,7]. Using these studies, under the conditions 2 2 0 u v uv    , (1 ) v  - constacyclic codes over 2 2 2

This report documents a five-step iterative experimental study aimed at enabling deterministic, property-based recall of knowledge-base entries from natural-language queries - building a Geometric Language Machine/Model.

Baseline diagnostics established that the incumbent SHA256-derived Golay encoding completely destroyed numeric proximity (mean Hamming distance ≈10.13/24 for near-identical inputs), making property-based retrieval impossible with the original pipeline. I designed and validated a Gray-code quantised numeric vector engine (117→142 bits) that preserves small numeric changes as small Hamming distances. Unified 166-bit representations (24 Golay + 142 Gray-code) were evaluated across four molecular representation strategies, revealing that stoichiometric additive composition fails for emergent molecular properties (e.g., water’s anomalously high boiling point). A five-step chat simulation demonstrated 7/7 targeted entries in the top-10 recall list across exact, partial, fuzzy, and qualitative query types, validating the proposed extension to the auto_trigger.py pipeline (not integrated into the UBP system).

The report concludes with a comprehensive discussion of successes, failures, and a roadmap for future development. This study was completed by K-Dense ai (as clearly branded) using my Universal Binary Principal (UBP) system.

In order to determine camera parameters, a calibration procedure involving the camera recordings of a checkerboard is usually performed. In this paper, we propose an alternative approach that uses Gray-code patterns displayed on an LCD screen. Gray-code patterns allow us to decode 3D location information of points of the LCD screen at every pixel in the camera image. This is in contrast to checkerboard patterns where the number of corresponding locations is limited to the number of checkerboard corners. We show that, for the case of a UEye CMOS camera, the precision of focal-length estimation is 1.5 times more precise than when using a standard calibration with a checkerboard pattern.

We propose new approaches to exploit compression algorithms for clustering numerical data. Our first contribution is to design a measure that can score the quality of a given clustering result under the light of a fixed encoding scheme. We call this measure the Minimum Code Length (MCL). Our second contribution is to propose a general strategy to translate any encoding method into a cluster algorithm, which we call COOL (COding-Oriented cLustering). COOL has a low computational cost since it scales linearly with the data set size. The clustering results of COOL is also shown to minimize MCL. To illustrate further this approach, we consider the Gray Code as the encoding scheme to present G-COOL. G-COOL can find clusters of arbitrary shapes and remove noise. Moreover, it is robust to change in the input parameters; it requires only two lower bounds for the number of clusters and the size of each cluster, whereas most algorithms for finding arbitrarily shaped clusters work well only if all parameters are tuned appropriately. G-COOL is theoretically shown to achieve internal cohesion and external isolation and is experimentally shown to work well for both synthetic and real data sets.

In this paper we proposed a new structure of GaAsP/6H-SiC/GaN Power semiconductor eld effect transistor with undoped region under gate. The device is made of semiconductor material i.e Gallium Nitride (GaN), Silicon Carbide and Gallium Arsenide Phospide and utilized Silvaco TCAD in 10nm technology for the simulation. The length and width of the undoped region is equal to length of channel and this undoped region under the gate will reduce the peak electric eld which existed in the channel region. The saturated drain current of proposed structure is about 53 percent higher than the conventional transistor and subsequently an improvement in other drain current parameters and analog parameters. The performance comparison is mainly in terms of drain current characteristics and analog characteristics such as Gain, transconductance, drain conductance, threshold voltage. As a subsequent improvement of performance parameters, this structure is used at high power and high frequency applications

In this paper, we investigate the lexicographic and colexicographic orderings of m-ary vectors of length n, as well as the mirror (left-recursive) reflected Gray code, complementing the classical m-ary reflected Gray code. We present efficient algorithms for generating vectors in each of these orders, each achieving constant amortized time per vector. Additionally, we propose algorithms implementing the four fundamental functions in generating combinatorial objects-successor, predecessor, rank, and unrank-each with linear time complexity. The properties and the relationships between these orderings and the set of integers {0, 1, . . . , m n -1} are examined in detail. We define explicit transformations between the different orders and illustrate them as a digraph very close to the complete symmetric digraph ← → K 5 . In this way, we provide a unified framework for understanding ranking, unranking, and order conversion. Our approach, based on emulating the execution of nested loops, proves powerful and flexible, leading to elegant and efficient algorithms that can be extended to the generation of submultisets, the generation of numbers in mixed-radix number systems, and related problems. The mirror m-ary Gray code introduced here has potential applications in coding theory and related areas.

Graphs and Algorithms A (cyclic) n-bit Gray code is a (cyclic) ordering of all 2(n) binary strings of length n such that consecutive strings differ in a single bit. Equivalently, an n-bit Gray code can be viewed as a Hamiltonian path of the n-dimensional hypercube Q(n), and a cyclic Gray code as a Hamiltonian cycle of Q(n). In this paper we study (cyclic) Gray codes avoiding a given set of faulty edges that form a matching. Given a matching M and two vertices u, v of Q(n), n >= 4, our main result provides a necessary and sufficient condition, expressed in terms of forbidden configurations for M, for the existence of a Gray code between u and v that avoids M. As a corollary. we obtain a similar characterization for a cyclic Gray code avoiding M. In particular, in the case that M is a perfect matching, Q(n) has a (cyclic) Gray code that avoids M if and only if Q(n) - M is a connected graph. This complements a recent result of Fink, who proved that every perfect matching of Q(n) can...

Various forms of Boolean minimization have been used within electronic engineering degrees as a key part of the syllabus. Typically, Karnaugh maps and Quine-McCluskey methods are the principal exhaustive search techniques for digital minimization at the undergraduate level as they are easy to use and simple to understand. Despite the popularity of these methods, they are not well suited to typical digital circuits. Simple examples of such circuits are parity, adders, gray code generators, and so on. The common factor among these is the Exclusive-Or logic gate. This problem is exacerbated by the increasing importance of Exclusive-Or in modern design. This paper proposes an extension to the Quine-McCIuskey method which successfully incorporates Exclusive-Or gates within the minimization process. A number of examples are given to demonstrate the effectiveness of this approach. This technique is easy to master as it can be considered to be an extension to the Quine-McCluskey method.

Hierarchical Memory Systems Necessity of implementing efficient y = Ax Sparse matrix Column ordering algorithms Experiments Conclusion and Future Work Performance gap between CPU speed and Main memory speed CPU speed improvement: 35% to 55% (in a year). Main memory latency improvement: 7% (in a year). Outline Hierarchical Memory Systems Necessity of implementing efficient y = Ax Sparse matrix Column ordering algorithms Experiments Conclusion and Future Work Thank you

The exact expression for the bit error rate (BER) of rectangular quadrature amplitude modulation (QAM) is given. The presented closed-form formula is independent of the bit mapping in use. It is thus particularly useful in the analysis of modulation schemes employing non-Gray mapping, e.g., those designed for iterative (turbo) demapping. Compared to the so-called expurgated bound and the union bound, our expression is shown to accurately predict the BER, which is particularly important in the low signal-to-noise ratio (SNR) range at which strong error-correcting codes ensure satisfactory performance.

Recent applications of quantum computing is the design and evaluations on the reversible logic technology. The digital systems widely utilize the code conversion technology in enhancing the data security and also minimize the required hardware. This nano-metric technology is high prolific in digital systems as it minimize the power consumption in logic circuits. This paper proposes the reversible logic design for such a code conversion including the binary to gray code, gray to binary code, BCD to excess 3 codes, excess 3 to BCD code.

Abstract: In this paper we briey introduce a Wide Spectrum Language and its transformation theory and describe a recent success of the theory: a general recursion removal theorem. This theorem includes as special cases the two techniques discussed by Knuth [12] and ...

An efficient semidefinite programming relaxation (SDPR) based virtually antipodal (VA) detection approach is proposed for Gray coded 16-QAM signalling over multiple-input-multiple-output (MIMO) channels. The existing index-bit-based VA-SDPR (IVA-SDPR) method is incapable of making direct binary decisions concerning the individual information bits without making symbol decisions first, except for the linear natural-mapping aided rectangular QAM constellations. By contrast, our new method is capable of directly deciding on the information bits of the ubiquitous Gray-mapping aided 16-QAM by employing a strikingly simple linear matrix representation (LMR) of 4-QAM. As an appealing benefit, the conventional "signal-to-symbol-to-bits" decision process is substituted by a simpler "signal-to-bits" decision process for the classic Gray-mapping aided rectangular 16-QAM. Furthermore, when combined with low-complexity bit-flipping based "hill climbing", the proposed direct-bit-based VA-SDPR (DVA-SDPR) detector achieves the best biterror-ratio (BER) performance among the known SDPR-based MIMO detectors in the context considered, while still maintaining a worst-case complexity order as low as O (4N T + 1) 3.5 .

Let G be a graph with diameter d. An antipodal labeling of G is a function f that assigns to each vertex a non-negative integer (label) such that for any two vertices u and v, it is satisfied that |f The antipodal number for G, denoted by an(G), is the minimum span of an antipodal labeling for G. Let C n denote the cycle on n vertices. Chartrand, Erwin and Zhang [4] determined the value of an(C n ) for n ≡ 2 (mod 4). In this article we determine the value of an(C n ) for n ≡ 1 (mod 4), confirming a conjecture of Chartrand el al. Moreover, we settle the case n ≡ 3 (mod 4); and improve the known lower bound and give an upper bound for the case n ≡ 0 (mod 4).

In this paper, we propose two algorithms for generating a complete n-bit binary reflected Gray code sequence. The first one is called Backtracking. It generates a complete n-bit binary reflected Gray code sequence by generating only a sub-tree instead of the complete tree. In this approach, both the sequence and its reflection for n-bit is generated by concatenating a "0" and a "1" to the most significant bit position of the n-1 bit result generated at each leaf node of the sub-tree. The second one is called MOptimal. It is the modification of a Space and time optimal approach [8] by considering the minimization of the total number of outer and inner loop execution for this purpose. In this method, both a sequence and its reflection for n-bit is also generated during each inner loop execution instead of generating only one sequence by adding "0" and "2n-1" to the previous n-1 bit sequence. Finally, we evaluate the performance of the proposed and existing algorithms by measuring the execution time with respect to number of bits and compare the results with each other. Simulation results demonstrate that Backtracking takes small execution time up to 10 bits with maximum improvement of 17.02%. The evaluation also demonstrates that MOptimal also takes small execution time for more than 15 bits showing an improvement of 2.10% for 23 bits.

This paper discusses reduction of the number of product terms in representation of totally symmetric Boolean functions by Sum of Products (SOP) and Fixed Polarity Reed-Muller (FPRM) expansions. The suggested method reduces the number of product terms, correspondingly, the implementation cost of symmetric functions based on these expressions by exploiting Gray decoding of input variables. Although this decoding is a particular example of all possible linear transformation of Boolean variables, it is efficient in the case of symmetric functions since it provides a significant simplification of SOPs and FPRMs. Mathematical analysis as well as experimental results demonstrate the efficiency of the proposed method.

The implementation cost of a multi-output Boolean function, in terms of the number of two-input AND-OR gates, can be reduced by using a linear decomposition. The linearly decomposed Boolean function consists of a linear function followed by the corresponding linearly transformed function. A complexity of the linearized function and therefore, its implementation cost, depends on the linear transform chosen. In this paper we suggest a spectral technique of the linear transformation of functions defined by disjoint cubes. The ...

Physical unclonable functions (PUFs) enable different characteristics according to the purpose, such as easy to access identification, high security level, and high code capacity, against counterfeiting a product. However, most multiplex approaches have been implemented by embedding several security features rather than one feature. In this paper, we present a high security level anti-counterfeiting strategy using only labyrinth wrinkle patterns with different complexities, which can be used as unique and unclonable codes. To generate codes with different levels in a microtaggant, we fabricated wrinkle patterns with characteristic wavelength gradients using grayscale lithography. The elastic modulus of the polymer substrate and corresponding wavelength after the wrinkling process were controlled by designing the gray level of each subcode region in a gray-level mask image for photopolymerization of the microparticle substrate. We then verified the uniqueness of the extracted minutia codes through a cross-correlation analysis. Finally, we demonstrated the authentication strategies by decoding different minutia codes according to the scanning resolution during the decoding. Overall, the presented patterning method can be widely used in security code generation.

Physical unclonable functions (PUFs) enable different characteristics according to the purpose, such as easy to access identification, high security level, and high code capacity, against counterfeiting a product. However, most multiplex approaches have been implemented by embedding several security features rather than one feature. In this paper, we present a high security level anti-counterfeiting strategy using only labyrinth wrinkle patterns with different complexities, which can be used as unique and unclonable codes. To generate codes with different levels in a microtaggant, we fabricated wrinkle patterns with characteristic wavelength gradients using grayscale lithography. The elastic modulus of the polymer substrate and corresponding wavelength after the wrinkling process were controlled by designing the gray level of each subcode region in a gray-level mask image for photopolymerization of the microparticle substrate. We then verified the uniqueness of the extracted minutia codes through a cross-correlation analysis. Finally, we demonstrated the authentication strategies by decoding different minutia codes according to the scanning resolution during the decoding. Overall, the presented patterning method can be widely used in security code generation.

Reliable image and video communications over noisy channels has been a great challenge especially for the transmission of large volume of data over unreliable and bandwidth limited channels. One technique to deal with this problem involves taking an existing source encoder and then protecting the bits using appropriate channel coding method. However, channel coding techniques add redundant bits resulting in an increase of the required bandwidth. Also, in the case of multimedia communications, the compressed source exhibits different level of importance among different portions of the encoded bit stream which calls for Unequal Error Protection (UEP). The unequal nature of bit importance that compressed source exhibits can be addressed by exploiting UEP within Hierarchical Quadrature Amplitude Modulation (HQAM) where different signalling points in a constellation are placed at different distances from the adjacent symbols according to their importance. This results in different bit er...

Gray-scale signals can be represented as sequences of integervalued symbols. If such a symbol has alphabet 0 1 . . . 2 1 it can be represented by binary digits. To embed information in these sequences, we are allowed to distort the symbols. The distortion measure that we consider here is squared error, however, errors larger than are not allowed. The embedded message must be recoverable with error probability zero. In this setup, there is a so-called "rate-distortion function" that tells us what the largest embedding rate is, given a certain distortion level and parameter . First, we determine this rate-distortion function for = 1 and for . Next we compare the performance of "low-bits modulation" to the rate-distortion function for . Then embedding codes are proposed based on i) ternary Hamming codes and on the ii) ternary Golay code. We show that all these codes are optimal in the sense that they achieve the smallest possible distortion at a given rate for fixed block length for any .

The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, large Schröder, Pell, even-index Fibonacci numbers and the central binomial coefficients. We thus provide Gray codes for the set of all permutations of {1, . . . , n} avoiding the pattern τ for all τ ∈ S 3 and the Gray codes we obtain have distances 4 or 5.

By using one-bit binary patterns instead of eight-bit sinusoidal ones, the binary defocusing techniques have been widely applied for high-speed 3D shape measurement. As projector defocusing is required, the phase unwrapping process of these techniques remains challenging. A recently proposed ternary Gray-code method can effectively increase the measurement speed by reducing the number of acquired coding patterns. However, it still has limitations, including the measuring range and a noise problem. To improve these, a new ternary Gray-code method is proposed by utilizing 2D modulation instead of the 1D modulation used in the conventional method. Simulations were conducted to compare these two methods, and the results verify that the proposed method reduces the variance in the introduced intermediate gray value by nearly 90%. Experiments were also carried out with the results verifying the superiority of the proposed method.

Fringe projection technologies have been widely used for three-dimensional (3D) shape measurement. One of the critical issues is absolute phase recovery, especially for measuring multiple isolated objects. This paper proposes a method for absolute phase retrieval using only one coded pattern. A total of four patterns including one coded pattern and three phase-shift patterns are projected, captured, and processed. The wrapped phase, as well as average intensity and intensity modulation, are calculated from three phase-shift patterns. A code word encrypted into the coded pattern can be calculated using the average intensity and intensity modulation. Based on geometric constraints of fringe projection system, the minimum fringe order map can be created, upon which the fringe order can be calculated from the code word. Compared with the conventional method, the measurement depth range is significantly improved. Finally, the wrapped phase can be unwrapped for absolute phase map. Since o...

Fringe projection systems have been widely applied in three-dimensional (3D) shape measurements. One of the important issues is how to retrieve the absolute phase. This paper presents a modified gray-level coding method for absolute phase retrieval. Specifically, two groups of fringe patterns are projected onto the measured objects, including three phase-shift patterns for the wrapped phase, and three n-ary gray-level (nGL) patterns for the fringe order. Compared with the binary gray-level (bGL) method which just uses two intensity values, the nGL method can generate many more unique codewords with multiple intensity values. With assistance from the average intensity and modulation of phase-shift patterns, the intensities of nGL patterns are normalized to deal with ambient light and surface contrast. To reduce the codeword detection errors caused by camera/projector defocus, nGL patterns are designed as n-ary gray-code (nGC) patterns to ensure that at most, one code changes at each ...

Genetic Algorithms comprise search and optimization strategies which are inspired by natural evolution: "survival of the fittest". In most of the best known basic genetic algorithms a binary coding of solution candidates is used. However for the DNA-coding, mother nature uses 4 purine bases: adenine (A), cytosine (C), guanine (G) and thymine (T). Following this idea, the present paper studies quaternary coded genetic algorithms and, based on high complex test functions, shows that these algorithms have a performance as good as their corresponding binary versions for problems with low dimensions and reach very fast an acceptable good fitness, if not the best, for high dimensional problems. For the first time it is shown, that the performance of genetic algorithms under a Gray code is sensitive to permutation of the columns of the code.

A well known problem of B. Grünbaum [Grü60] asks whether for every continuous mass distribution (measure) dµ = f dm on R n there exist n hyperplanes dividing R n into 2 n parts of equal measure. It is known that the answer is positive in dimension n = 3 [Had66] and negative for n ≥ 5, [Avis84] [Ram96]. We give a partial solution to Grünbaum's problem in the critical dimension n = 4 by proving that each measure µ in R 4 admits an equipartition by 4 hyperplanes, provided that it is symmetric with respect to a 2-dimensional, affine subspace L of R 4 . Moreover we show, by computing the complete obstruction in the relevant group of normal bordisms, that without the symmetry condition, a naturally associated topological problem has a negative solution. The computation is based on the Koschorke's exact singularity sequence [Kosch81] and the remarkable properties of the essentially unique, balanced, binary Gray code in dimension 4, [Toot56] [Knu01].

The past decade has seen a flurry of research into pattern avoiding permutations but little of it is concerned with their exhaustive generation. Many applications call for exhaustive generation of permutations subject to various constraints or imposing a particular generating order. In this paper we present generating algorithms and combinatorial Gray codes for several families of pattern avoiding permutations. Among the families under consideration are those counted by Catalan, large Schröder, Pell, even-index Fibonacci numbers and the central binomial coefficients. We thus provide Gray codes for the set of all permutations of {1, . . . , n} avoiding the pattern τ for all τ ∈ S 3 and the Gray codes we obtain have distances 4 or 5.

In this paper we introduce a family of filter kernelsthe Gray-Code Kernels (GCK) and demonstrate their use in image analysis. Filtering an image with a sequence of Gray-Code Kernels is highly efficient and requires only 2 operations per pixel for each filter kernel, independent of the size or dimension of the kernel. We show that the family of kernels is large and includes the Walsh-Hadamard kernels amongst others. The GCK can also be used to approximate arbitrary kernels since a sequence of GCK can form a complete representation. The efficiency of computation using a sequence of GCK filters can be exploited for various real-time applications, such as, pattern detection, feature extraction, texture analysis, and more.

In this paper, we introduce a family of filter kernels-the Gray-Code Kernels (GCK) and demonstrate their use in image analysis. Filtering an image with a sequence of Gray-Code Kernels is highly efficient and requires only two operations per pixel for each filter kernel, independent of the size or dimension of the kernel. We show that the family of kernels is large and includes the Walsh-Hadamard kernels, among others. The GCK can be used to approximate any desired kernel and, as such forms, a complete representation. The efficiency of computation using a sequence of GCK filters can be exploited for various real-time applications, such as, pattern detection, feature extraction, texture analysis, texture synthesis, and more.

Motion estimation plays an important role in modern video coders. In such coders, motion is estimated using a block matching algorithm that estimates the amount of motion on a blockby-block basis. A full search technique for finding the best matching blocks delivers good accuracy but is usually not practical due to its high computational complexity. In this paper, a novel fast blockbased motion estimation algorithm is proposed. This algorithm uses an efficient projection framework which bounds the distance between a template block and candidate blocks. Fast projection is performed with a family of highly efficient filter kernels -the Gray Code Kernels -using only 2 operations per pixel for each filter kernel. The projection framework is combined with a rejection scheme which allows rapid rejection of candidate blocks that are distant from the template block. Experiments show that the proposed algorithm significantly outperforms popular fast motion estimation algorithms, such as three-step search and diamond search. In addition, the tradeoff between computational complexity and quality of results could be easily controlled in the proposed algorithm, thus it enables adaptivity to image content.

Based on BRGC inspired order relations we define Gray codes and give a generating algorithm for q-ary words avoiding a prescribed factor. These generalize an early 2001 result and a very recent one published by some of the present authors, and can be seen as an alternative to those of Squire published in 1996. Among the involved tools, we make use of generalized BRGC order relations, ultimate periodicity of infinite words, and word matching techniques.

We provide a trace partitioned Gray code for the set of q-ary strings avoiding a pattern constituted by k consecutive equal symbols. The definition of this Gray code is based on two different constructions, according to the parity of q. This result generalizes, and is based on, a Gray code for binary strings avoiding k consecutive 0's.

Based on BRGC inspired order relations we define Gray codes and give a generating algorithm for q-ary words avoiding a prescribed factor. These generalize an early 2001 result and a very recent one published by some of the present authors, and can be seen as an alternative to those of Squire published in 1996. Among the involved tools, we make use of generalized BRGC order relations, ultimate periodicity of infinite words, and word matching techniques.

A cross-bifix-free set of words is a set in which no prefix of any length of any word is the suffix of any other word in the set. A construction of cross-bifix-free sets has recently been proposed in Cheeet al.(2013) within a constant factor of optimality. We propose a Gray code for these cross-bifix-free sets and a CAT algorithm generating it. Our Gray code list is trace partitioned, that is, words with zero in the same positions are consecutive in the list.

Image encryption is an efficient and vital way to protect classified and secret images. With the advancement of the processing power of the computer, AES, DES, or chaotic series type just alike image encryption schemes are not as secure as before. Therefore, in this paper, we present a new hybrid image encryption method for protecting secret and imperative images by employing logistic sine system (LSS) together with two-dimensional cellular automata and FSM-based DNA rule generator. The secure hash (SHA-256) algorithm is used to generate a secret key and to compute initial values for the LSS. In our proposed method, there are three stages and each stage has its own rule. After the scrambling process, the first stage is the Feistel structure-based bit inversion (FSBI) to change the pixels' value. The second stage is 2D-CA with Moore neighborhood structure-based local rules. The third is DNA conversion based on finite-state machine (FSM-DNA) rule generator. The proposed encryption scheme is robust against the well-known attacks, such as statistical attacks, brute force attacks, differential attacks, and pixel correlation attacks, and also possesses strong key sensitivity. The results show that our three-layer hybrid image encryption technique is robust against many well-known attacks and can be applied directly to all types of classified gray-scale images to make them more secure from such cryptography attacks.

In this paper, we develop a constant amortized time (CAT) algorithm for generating permutations with a fixed number of excedances. We obtain a Gray code for permutations having one excedance. We also give a bijection between the set of n-length permutations with exactly one excedance and the set Sn(321, 2413, 3412, 21534)\{123 . . . (n -1)n}. This induces a Gray code for the set S n (321, 2413, 3412, 21534)\{123 . . . (n -1)n}.

In , the author provided a Gray code for the set of n-length permutations with a given number of left-to-right minima in inversion array representation. In this paper, we give the first Gray code for the set of n-length permutations with a given number of left-to-right minima in one-line representation. In this code, each permutation is transformed into its successor by a product with a transposition or a cycle of length three. Also a generating algorithm for this code is given.

We give a Gray code and constant average time generating algorithm for derangements, i.e., permutations with no ÿxed points. In our Gray code, each derangement is transformed into its successor either via one or two transpositions or a rotation of three elements. We generalize these results to permutations with number of ÿxed points bounded between two constants.

A new combinatorial characterization of a gray-tone image called Euler Vector is proposed. The Euler number of a binary image is a well-known topological feature, which remains invariant under translation, rotation, scaling, and rubber-sheet transformation of the image. The Euler vector comprises a 4-tuple, where each element is an integer representing the Euler number of the partial binary image formed by the gray-code representation of the four most significant bit planes of the gray-tone image. Computation of Euler vector requires only integer and Boolean operations. The Euler vector is experimentally observed to be robust against noise and compression. For efficient image indexing, storage and retrieval from an image database using this vector, a bucket searching technique based on a simple modification of Kd-tree, is employed successfully. The Euler vector can also be used to perform an efficient four-dimensional range query. The set of retrieved images are finally ranked on the basis of Mahalanobis distance measure. Experiments are performed on the COIL database and results are reported. The retrieval success can be improved significantly by augmentiong the Euler vector by a few additional simple shape features. Since Euler vector can be computed very fast, the proposed technique is likely to find many applications to content-based image retrieval.

In the contemporary era, safeguarding vast amounts of sensitive data and ensuring secure communication are paramount concerns, especially given the prevalence of insecure networks. The fields of cryptography and steganography have emerged as pivotal tools in addressing these challenges, attracting significant scientific attention due to their established effectiveness. This paper proposes a sophisticated double-layered security system for protecting textual data, initially utilizing 2-D Cellular automata (CA) to establish robust encryption protocols. The incorporation of the Von Neumann neighborhood of 2-D CA enhances the generation of secure cryptographic keys, leveraging its complex and chaotic behavior to improve randomness and unpredictability. Subsequently, to enhance security, the discrete cosine transform (DCT) technique is integrated, facilitating discreet embedding of encrypted data into the least significant bit (LSB) of DCT coefficients. An exhaustive range of tests and evaluations, encompassing analysis of key space, complexity, performance metrics, avalanche effect, security attacks, peak signal-to-noise ratio (PSNR) and mean square error (MSE), substantiates the effectiveness and resilience of this multifaceted approach. The findings reveal a significant average avalanche effect of 76%, indicating resilience against cryptographic attacks. Furthermore, superior preservation of image quality is observed during encryption, as evidenced by improved PSNR and MSE values compared to alternative technique. The overall analyses validate the proposed method as a proficient solution for secure data encryption in cyberspace.

We examine the problem of integer representation in near minimal number of bits so that increment and decrement (and indeed addition and subtraction) can be performed using few bit inspections and fewer bit changes. In particular, we prove a new lower bound of Ω(√ n) for the increment and decrement operation, where n is the minimum number of bits required to represent the number. The model of computation we considered is the bit probe model, where the complexity measure counts only the bitwise accesses to the data structure. We present several efficient data structures to represent integer that use a logarithmic number of bit inspections and a constant number of bit changes per operation.