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The aim of this work is to obtain two types of regularities for calculating the number of palindromes. The main results of the work are presented by solving two problems. In the first problem describes how to find the total number of different locations of palindromes in binary digits that do not exceed the given numbers of digits. The second problem is devoted to determining the number of palindromes of an arbitrary number system that have the same numbers of digits. The theoretical results of the work are accompanied by examples of the application of the described regularities to obtain the total number of locations of the binary palindromes with the number of digits that do not exceed the specified, as well as to calculate the number of palindromes of different number systems with predetermined numbers of digits. The considered regularities are supposed to be applied in the analysis of numerical and computer data.

While Shannon introduced “binary digits, or more briefly bits” as units of information, a binary digit (and, thus, a bit) originally and still means a character of “0” or “1”, which may be represented by a binary symbol or storage cell. As a result, a bit often means a binary symbol or storage cell, which is inherited by a “qubit”, a quantum bit, introduced by Schumacher as “the fundamental units of quantum information”. Schumacher himself uses a bit, not a qubit, as a unit of quantum information and a qubit is a quantum binary symbol or storage cell.

For example G(V,E) becomes graph on p nodes and q sides and B'(G) called being side-node incidence matrix from G. Every side e of G can be labeled using a binary digit string of length n from the row of B'(G) which corresponds to the side e, denoted by s(e). The Hamming distance between the strings s(e) and s(f) of length n is defined to be the number of positions of s(e) and s(f) with different digit. Hamming index from a graph is the number of Hamming distances between all pairs of strings. In this research, author discuss the Hamming index of graphs produced by an side-node incidence matrix, particularly windmill graph and snake graphs.

Anticounterfeiting tags affixed to products offer a practical solution to combat counterfeiting. To be effective, these tags must be economical, capable of ultrafast production, mass-producible, easy to authenticate, and automatable. We present a universal laser ablation technique that rapidly generates intrinsic, randomly distributed craters (in under a second) on laser-sensitive materials using a nanosecond pulsed infrared laser. The laser and scanning line parameters are balanced to produce randomly distributed craters. The tag patterns demonstrate high randomness, which is analyzed using pattern recognition algorithms and root mean square error deviation. The optical image information of the tag is digitized with a fixed bit uniformity of 0.5 without employing any debiasing algorithm. The efficacy of tags for anticounterfeiting is presented by securing the challenge associated with each tag. Statistical NIST tests are successfully performed on responses generated from both single and multiple tags, demonstrating the true randomness of the sequence of binary digits. The single(multiple) tag(s) achieved an actual encoding capacity of approximately 10 391 (10 518 ) and a low false rate (both positive and negative) on the order of 10 −58 (10 −50 ). Our findings introduce a laser-based method for anticounterfeiting tag generation, allowing for ultrafast and straightforward product processing with minimal fabrication and tag cost. Methods to realise anticounterfeiting labels should be fast, easy to implement, cheap, and applicable to several different substrates. Here, the authors demonstrate how to use laser ablation to produce randomly distributed craters that can be used as anticounterfeiting tags.

A DNA barcode is a short piece of standard DNA sequence used for species determination and discrimination. Representation of DNA barcodes is essential for DNA barcodes’ applications in the transportation and recognition of biological materials. Previously, we have compared different strategies for representing the DNA barcodes. In the present study, we have developed a compression algorithm based on binary coding or Huffman coding scheme, followed by converting the binary digits into Base64 digits. The combination of this compression algorithm and the QR representation leads to the dynamic DNA QR coding algorithm (DDQR). We tested the DDQR algorithm on simulated data and real DNA barcode sequences from the commonly used plant and animal DNA barcode markers: rbcL, matK, trnH-psbA, ITS2, and COI. We compared the compression efficiency of DDQR and another state-of-the-art DNA compression algorithm GeCo3 for sequences with various base compositions and lengths. We found that DDQR had a higher compression rate than GeCo3 for DNA sequences shorter than 800 bp, which is the typical size range for DNA barcodes. We also upgraded a web server ( http://www.1kmpg.cn/ddqr ) that provides three functions: retrieval of DNA barcode sequences, encoding DNA barcode sequences to DDQR codes, and decoding DDQR codes to DNA barcode sequences. The DDQR algorithm and the webserver will be invaluable to applying DNA barcode technology in the food and traditional medicine industries.

Secured Compressive sensing of the images becomes one of the essential issues in multimedia applications. In the recent times, encryption in compressive sensing is achieved via the use of multiple one dimensional chaotic system (1D chaotic) and hyper-chaotic system (HC). However, the security of the system still needs to be improved. To solve this issue, novel encrypted compressive sensing of images based on Fractional order hyper chaotic Chen system and DNA operations is proposed in this paper. The basic idea is to introduce a new algorithm which provides compression rate below the Nyquist rate along with increased security by jointly using Fractional order hyper chaotic chen system and DNA operations for image encryption. The encryption algorithm combines Fractional order hyperchaotic chen system and DNA operations. Fractional order hyperchaotic chen system has high randomness and rich dynamic phenomena which enhance the encryption and the algorithm efficiency is further increased by DNA operations.4-Dimensioanal Fractional order hyper chaotic chen system is used to generate the measurement matrix. Compressive sensing measurements are converted to a stream of binary digits and the correlation between the adjacent bits is further reduced through global scrambling. The DNA operations are performed on the scrambled binary sequences and hyper chaotic sequences, which increases the algorithm efficiency. The results of the experiments accomplish that the proposed encryption method is extremely sensitive to small changes in secret keys, shows good performance in histogram analysis, correlation analysis, Information Entropy and is very sensitive to a bit change in an input image. The Block Compressive sensing (BCS) reconstruction algorithms are used to validate the proposed encryption method. In the proposed method, experimental analysis are performed on different images of size 512 × 512 which are divided in to blocks of size 32 X 32. The reconstruction analysis is performed with different subsampling rates of 0.1, 0.2, 0.3, 0.4 and 0.5.The results depicted that the proposed method maintains the robustness and reconstruction quality of Compressive sensing with enhanced encryption.

We use bounds of character sums and some combinatorial arguments to show the abundance of very smooth numbers which also have very few nonzero binary digits.

The paper considers an approach to comparing the number modules represented in the number systems in residual classes (RNS), one of the bases of which is pn = 2k, where k = 2, 3, 4, ... The approach involves the following sequence of actions. The decrease in number modules |A| and |B| by αn and ßn, respectively, where αn = rest|A| mod pn and ßn = rest|B| mod pn. Next, access to the computer memory at the addresses (α1, α2, . . ., αn-1) and (ß1, ß2, . . . ., ßn-1) and selection from the memory the high digits (without k low digits) of the modules |А| and |В| represented in the positional binary system by comparing the selected high digits of the modules. In this case, a larger module will correspond to larger high digits. If the high digits of the modules are equal then the lower digits are compared which coincide with the residues αn and ßn. In this case, the largest of the lower digits will correspond to the larger module. With this approach, the memory required to store the compared modules when they are written in the positional binary number system is reduced by 2k times, and the word length of the stored words decreases by k binary digits. In addition, the low bit depth of the RNS bases allows the using of tabular calculation methods which increases the speed of calculations. Thus the proposed approach has a practical orientation and may be of interest to computer developers.

We prove a folklore conjecture concerning the sum-of-digits functions in bases two and three: there are infinitely many positive integers n such that the sum of the binary digits of n equals the sum of the ternary digits of n .

Gain of multi-relay coding cooperation. As the number of joint iterations increases, the bit error rate performance increases rapidly, which is significantly better than the coding non-cooperative system under the same conditions. The source output should be represented with as few binary digits as possible. Various noises and interferences in the channel are the main causes of bit errors in digital communication system receivers. Collaborative technology can improve the performance of the system without significantly increasing the system bandwidth or increasing the transmit power, and has attracted more and more attention in the academic and engineering fields.