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Journal articles on the topic 'Code distance'

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Published: 30 May 2022

Last updated: 31 May 2022

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1

Gabidulin, E. M., and N. I. Pilipchuk. "Multicomponent codes with maximum code distance." Problems of Information Transmission 52, no. 3 (July 2016): 276–83. http://dx.doi.org/10.1134/s0032946016030054.

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2

Soloviev, Alexander A., and Dmitry V. Chernikov. "Biorthogonal wavelet codes with prescribed code distance." Discrete Mathematics and Applications 28, no. 3 (June 26, 2018): 179–88. http://dx.doi.org/10.1515/dma-2018-0017.

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Abstract We propose a scheme of construction of 2-circulant codes with given code distance on the basis of biorthogonal filters with the property of perfect reconstruction over a finite filed of odd characteristic. The corresponding algorithm for constructing biorthogonal filters utilizes the Euclidean algorithm for finding the gcd of polynomials.

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3

Stepanov, S. A. "Nonlinear q-ary codes with large code distance." Problems of Information Transmission 53, no. 3 (April 4, 2017): 242–50. http://dx.doi.org/10.1134/s003294601703005x.

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4

Hall, J. I., and J. H. van Lint. "Constant distance code pairs." Indagationes Mathematicae (Proceedings) 88, no. 1 (March 1985): 41–45. http://dx.doi.org/10.1016/s1385-7258(85)80018-4.

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5

Olivares, J., L. M. Sarro, H. Bouy, N. Miret-Roig, L. Casamiquela, P. A. B. Galli, A. Berihuete, and Y. Tarricq. "Kalkayotl: A cluster distance inference code." Astronomy & Astrophysics 644 (November 24, 2020): A7. http://dx.doi.org/10.1051/0004-6361/202037846.

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Context. The high-precision parallax data of the Gaia mission allows for significant improvements in the distance determination to stellar clusters and their stars. In order to obtain accurate and precise distance determinations, systematics such as parallax spatial correlations need to be accounted for, especially with regard to stars in small sky regions. Aims. Our aim is to provide the astrophysical community with a free and open code designed to simultaneously infer cluster parameters (i.e., distance and size) and distances to the cluster stars using Gaia parallax measurements. The code includes cluster-oriented prior families and it is specifically designed to deal with the Gaia parallax spatial correlations. Methods. A Bayesian hierarchical model is created to allow for the inference of both the cluster parameters and distances to its stars. Results. Using synthetic data that mimics Gaia parallax uncertainties and spatial correlations, we observe that our cluster-oriented prior families result in distance estimates with smaller errors than those obtained with an exponentially decreasing space density prior. In addition, the treatment of the parallax spatial correlations minimizes errors in the estimated cluster size and stellar distances, and avoids the underestimation of uncertainties. Although neglecting the parallax spatial correlations has no impact on the accuracy of cluster distance determinations, it underestimates the uncertainties and may result in measurements that are incompatible with the true value (i.e., falling beyond the 2σ uncertainties). Conclusions. The combination of prior knowledge with the treatment of Gaia parallax spatial correlations produces accurate (error < 10%) and trustworthy estimates (i.e., true values contained within the 2σ uncertainties) of cluster distances for clusters up to ∼5 kpc, along with cluster sizes for clusters up to ∼1 kpc.

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6

Delfosse, Nicolas, and Matthew B. Hastings. "Union-Find Decoders For Homological Product Codes." Quantum 5 (March 10, 2021): 406. http://dx.doi.org/10.22331/q-2021-03-10-406.

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Homological product codes are a class of codes that can have improved distance while retaining relatively low stabilizer weight. We show how to build union-find decoders for these codes, using a union-find decoder for one of the codes in the product and a brute force decoder for the other code. We apply this construction to the specific case of the product of a surface code with a small code such as a [[4,2,2]] code, which we call an augmented surface code. The distance of the augmented surface code is the product of the distance of the surface code with that of the small code, and the union-find decoder, with slight modifications, can decode errors up to half the distance. We present numerical simulations, showing that while the threshold of these augmented codes is lower than that of the surface code, the low noise performance is improved.

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7

Hastings, Mathew B. "Weight reduction for quantum codes." Quantum Information and Computation 17, no. 15&16 (December 2017): 1307–34. http://dx.doi.org/10.26421/qic17.15-16-4.

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We present an algorithm that takes a CSS stabilizer code as input, and outputs another CSS stabilizer code such that the stabilizer generators all have weights O(1) and such that O(1) generators act on any given qubit. The number of logical qubits is unchanged by the procedure, while we give bounds on the increase in number of physical qubits and in the effect on distance and other code parameters, such as soundness (as a locally testable code) and “cosoundness” (defined later). Applications are discussed, including to codes from high-dimensional manifolds which have logarithmic weight stabilizers. Assuming a conjecture in geometry[11], this allows the construction of CSS stabilizer codes with generator weight O(1) and almost linear distance. Another application of the construction is to increasing the distance to X or Z errors, whichever is smaller, so that the two distances are equal.

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8

Zhu, Bing. "Rethinking Fractional Repetition Codes: New Construction and Code Distance." IEEE Communications Letters 20, no. 2 (February 2016): 220–23. http://dx.doi.org/10.1109/lcomm.2015.2512871.

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9

Noguchi, Satoshi, Xiao-Nan Lu, Masakazu Jimbo, and Ying Miao. "BCH Codes with Minimum Distance Proportional to Code Length." SIAM Journal on Discrete Mathematics 35, no. 1 (January 2021): 179–93. http://dx.doi.org/10.1137/19m1260876.

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10

Talmale, Seema, Srija Unnikrishnan, and Bhaurao K. Lande. "Distance increasing mapping for variable distance block code." IET Communications 14, no. 9 (June 2, 2020): 1495–501. http://dx.doi.org/10.1049/iet-com.2019.0875.

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11

Heiss, M. "Error-detecting unit-distance code." IEEE Transactions on Instrumentation and Measurement 39, no. 5 (1990): 730–34. http://dx.doi.org/10.1109/19.58616.

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12

LINT, J. H. VAN. "Distance Theorems for Code Pairs." Annals of the New York Academy of Sciences 555, no. 1 Combinatorial (May 1989): 421–24. http://dx.doi.org/10.1111/j.1749-6632.1989.tb22481.x.

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13

Hastings, Mathew B. "Small Majorana fermion codes." Quantum Information and Computation 17, no. 13&14 (November 2017): 1191–205. http://dx.doi.org/10.26421/qic17.13-14-7.

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We consider Majorana fermion stabilizer codes with small number of modes and distance. We give an upper bound on the number of logical qubits for distance 4 codes, and we construct Majorana fermion codes similar to the classical Hamming code that saturate this bound. We perform numerical studies and find other distance 4 and 6 codes that we conjecture have the largest possible number of logical qubits for the given number of physical Majorana modes. Some of these codes have more logical qubits than any Majorana fermion code derived from a qubit stabilizer code.

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14

GUERRINI, ELEONORA, EMMANUELA ORSINI, and MASSIMILIANO SALA. "COMPUTING THE DISTANCE DISTRIBUTION OF SYSTEMATIC NONLINEAR CODES." Journal of Algebra and Its Applications 09, no. 02 (April 2010): 241–56. http://dx.doi.org/10.1142/s0219498810003884.

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The most important families of nonlinear codes are systematic. A brute-force check is the only known method to compute their weight distribution and distance distribution. On the other hand, it outputs also all closest word pairs in the code. In the black-box complexity model, the check is optimal among closest-pair algorithms. In this paper, we provide a Gröbner basis technique to compute the weight/distance distribution of any systematic nonlinear code. Also our technique outputs all closest pairs. Unlike the check, our method can be extended to work on code families.

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15

Farrelly, Terry, David K. Tuckett, and Thomas M. Stace. "Local tensor-network codes." New Journal of Physics 24, no. 4 (April 1, 2022): 043015. http://dx.doi.org/10.1088/1367-2630/ac5e87.

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Abstract Tensor-network codes enable the construction of large stabilizer codes out of tensors describing smaller stabilizer codes. An application of tensor-network codes was an efficient and exact decoder for holographic codes. Here, we show how to write some topological codes, including the surface code and colour code, as simple tensor-network codes. We also show how to calculate distances of stabilizer codes by contracting a tensor network. The algorithm actually gives more information, including a histogram of all logical coset weights. We prove that this method is efficient in the case of stabilizer codes encoded via local log-depth circuits in one dimension and holographic codes. Using our tensor-network distance calculator, we find a modification of the rotated surface code that has the same distance but fewer minimum-weight logical operators by ‘doping’ the tensor network, i.e., we break the homogeneity of the tensor network by locally replacing tensors. For this example, this corresponds to an improvement in successful error correction of almost 2% against depolarizing noise (in the perfect-measurement setting), but comes at the cost of introducing three higher-weight stabilizers. Our general construction lets us pick a network geometry (e.g., a Euclidean lattice in the case of the surface code), and, using only a small set of seed codes (constituent tensors), build extensive codes with the potential for optimisation.

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16

Salnikov, O. N. "The analysis of LDPC code rate and traditional antinoise codes." Informacionno-technologicheskij vestnik 14, no. 4 (December 30, 2017): 87–90. http://dx.doi.org/10.21499/2409-1650-2017-4-87-90.

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17

Frenger, P., P. Orten, and T. Ottosson. "Code-spread CDMA using maximum free distance low-rate convolutional codes." IEEE Transactions on Communications 48, no. 1 (2000): 135–44. http://dx.doi.org/10.1109/26.818881.

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18

Yegorov, S. I., D. B. Borzov, S. V. Degtyarev, V. A. Dreizin, and I. B. Mikhailov. "INCREASE IN EFFICIENCY OF DECODING OF CODES OF READ-SOLOMON ON THE GENERALIZED MINIMUM DISTANCE." Proceedings of the Southwest State University 22, no. 3 (June 28, 2018): 51–58. http://dx.doi.org/10.21869/2223-1560-2018-22-3-51-58.

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In the modern systems of transfer and storage of information for correction of the arising mistakes noiseproof codes of Read-Solomon widely are used. With use of soft decisions apply decoding of these codes on the generalized minimum distance which advantage is simplicity of realization to correction of mistakes. In work the algorithm of decoding of codes of Read-Solomon on the generalized minimum distance which feature is use of the algebraic decoder correcting errors abroad a half of the minimum code distance with use of soft decisions is offered. The algebraic decoder realizes syndromic decoding and is based on application of analytical continuation of an algorithm of Berlekempa-Messi for 2τ iterations (τ-number of in addition corrected wrong symbols). He provides search of positions of tC+τ of wrong symbols in a code word (tC - number of the wrong symbols which are guaranteed corrected by a code) which locators would be the return to roots of a possible polynom of locators of errors of degree tC + τ. Search of positions of mistakes is carried out in ascending order of nadezhnost of symbols of the accepted code word. The efficiency of correction of mistakes was investigated by the offered algorithm in the channel with additive white Gaussian noise by imitating modeling on the COMPUTER. Researches were conducted for Read-Solomon's codes defined over the field of GF(28). The additional code prize provided with an algorithm at correction on iteration of three additional mistakes in relation to Read-Solomon (255,239,17) code reaches 0,26 dB. The additional code prize for Read-Solomon (255,127,129) code at correction on iteration of two additional mistakes has made about 0,1 dB. The additional code prize for Read-Solomon (255,41,215) code at correction on iteration of three additional mistakes has made about 0,17 dB.

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19

Martinyuk, T. B., O. V. Wojciechowska, and O. S. Gorodets. "Equidistance and unit codes." Optoelectronic Information-Power Technologies 41, no. 1 (May 2, 2022): 20–24. http://dx.doi.org/10.31649/1681-7893-2021-41-1-20-24.

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In coding theory, single codes, as the implementation of non-traditional coding methods, are focused in particular on such a practical application as the transmission of symbols (bits) in communication channels. Today, the analysis and study of unit codes is a priority and actuality in terms of evaluating their corrective properties. This paper analyzes the properties of three unit codes, such as unit position (marking), unit pair and unit normal codes. The comparative characteristic of these codes taking into account their correcting properties is given. It is taken into account that the corrective properties of the code are determined by the code distance, which is the minimum distance between its code points. The formulas for determining the average probability of error non-detection for the received correction codes are given. Research and proof of corrective properties of unit codes were carried out from the point of view of their equidistance. Such codes are characterized by the fact that the code distance in the equidistant code must be an even number. The analysis of the characteristics of unit codes presented in this work showed that one of the optimal among equidistant unit codes can be considered a unit position (marking) code. The formula for calculating the lower estimate of the average probability of error non-detection for any probability of error-free transmission of the symbol, which coincides with the value of the average probability of error non-detection for the McDonald's equidistant code. This also confirmed the optimality of a considered unit position (marking) code. With this in mind, the application area of the unit position code, as noise immunity, extends due to the possibility of encoding the states of correcting machines and addressing data in computer storage devices.

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20

Guo, Luobin, Qiang Fu, Ruihu Li, and Liangdong Lu. "Maximal entanglement entanglement-assisted quantum codes of distance three." International Journal of Quantum Information 13, no. 01 (February 2015): 1550002. http://dx.doi.org/10.1142/s0219749915500021.

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Entanglement-assisted quantum error correcting code (EAQECC) is a generalization of standard stabilizer quantum code. Maximal entanglement EAQECCs can achieve the EA-hashing bound asymptotically. In this work, we give elementary recursive constructions of quaternary zero radical codes with dual distance three for all n ≥ 4. Consequently, good maximal entanglement EAQECCs of minimum distance three for such length n are obtained. Almost all of these EAQECCs are optimal or near optimal according to the EA-quantum Hamming bound.

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21

Hartanto, Ari Dwi, and Al Sutjijana. "Binary Cyclic Pearson Codes." Jurnal Matematika MANTIK 7, no. 1 (March 18, 2021): 1–8. http://dx.doi.org/10.15642/mantik.2021.7.1.1-8.

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The phenomena of unknown gain or offset on communication systems and modern storages such as optical data storage and non-volatile memory (flash) becomes a serious problem. This problem can be handled by Pearson distance applied to the detector because it offers immunity to gain and offset mismatch. This distance can only be used for a specific set of codewords, called Pearson codes. An interesting example of Pearson code can be found in T-constrained code class. In this paper, we present binary 2-constrained codes with cyclic property. The construction of this code is adopted from cyclic codes, but it cannot be considered as cyclic codes.

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22

Olshevska, V. A. "Permutation codes over Sylow 2-subgroups $Syl_2(S_{2^n})$ of symmetric groups $S_{2^n}$." Researches in Mathematics 29, no. 2 (December 30, 2021): 28. http://dx.doi.org/10.15421/242107.

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The permutation code (or the code) is well known object of research starting from 1970s. The code and its properties is used in different algorithmic domains such as error-correction, computer search, etc. It can be defined as follows: the set of permutations with the minimum distance between every pair of them. The considered distance can be different. In general, there are studied codes with Hamming, Ulam, Levensteins, etc. distances.In the paper we considered permutations codes over 2-Sylow subgroups of symmetric groups with Hamming distance over them. For this approach representation of permutations by rooted labeled binary trees is used. This representation was introduced in the previous author's paper. We also study the property of the Hamming distance defined on permutations from Sylow 2-subgroup $Syl_2(S_{2^n})$ of symmetric group $S_{2^n}$ and describe an algorithm for finding the Hamming distance over elements from Sylow 2-subgroup of the symmetric group with complexity $O(2^n)$. The metric properties of the codes that are defined on permutations from Sylow 2-subgroup $Syl_2(S_{2^n})$ of symmetric group $S_{2^n}$ are studied. The capacity and number of codes for the maximum and the minimum non-trivial distance over codes are characterized.

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23

Fan, Di, Jia Li, Mao Yong Cao, Nong Liang Sun, and Hong Yu. "Recognizing Check Magnetic Code Based on Peak-Valley Code and Distance." Applied Mechanics and Materials 145 (December 2011): 588–92. http://dx.doi.org/10.4028/www.scientific.net/amm.145.588.

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The check is a popular form for the non-cash payment and accounts for more than 50% of the non-cash transactions. Magnetic ink character recognition system can recognize the check magnetic code automatically and get the information of the bank and account. In magnetic ink character recognition system, the recognizing algorithm is mostly based on correlation coefficient. The computational cost of this algorithm is very high. This paper has proposed a new algorithm based on the peak-valley code and peak-valley distance after analyzing the characteristics of magnetic code signals in E-13B standards to simplify the calculations and system design. Firstly, the magnetic code signal is normalized and separated into magnetic character signals by the thresholds of peak and valley. Secondly, the features of the peak-valley code and peak-valley distance of each magnetic character signal are extracted, then the recognition based on peak-valley code and the nearest neighbor recognition algorithm based on peak-valley distance are utilized to recognize the magnetic code. The recognition results and statistical parameters from a large number of experiments show that the new method has high recognition rate, good robustness and low computational cost.

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24

Chamberland, Christopher, and Michael E. Beverland. "Flag fault-tolerant error correction with arbitrary distance codes." Quantum 2 (February 8, 2018): 53. http://dx.doi.org/10.22331/q-2018-02-08-53.

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In this paper we introduce a general fault-tolerant quantum error correction protocol using flag circuits for measuring stabilizers of arbitrary distance codes. In addition to extending flag error correction beyond distance-three codes for the first time, our protocol also applies to a broader class of distance-three codes than was previously known. Flag circuits use extra ancilla qubits to signal when errors resulting fromvfaults in the circuit have weight greater thanv. The flag error correction protocol is applicable to stabilizer codes of arbitrary distance which satisfy a set of conditions and uses fewer qubits than other schemes such as Shor, Steane and Knill error correction. We give examples of infinite code families which satisfy these conditions and analyze the behaviour of distance-three and -five examples numerically. Requiring fewer resources than Shor error correction, flag error correction could potentially be used in low-overhead fault-tolerant error correction protocols using low density parity check quantum codes of large code length.

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25

Çalkavur, Selda. "Public-Key Cryptosystems and Bounded Distance Decoding of Linear Codes." Entropy 24, no. 4 (April 1, 2022): 498. http://dx.doi.org/10.3390/e24040498.

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Error-correcting codes form an important topic in information theory. They are used to correct errors that occur during transmission on a noisy channel. An important method for correcting errors is bounded distance decoding. The public-key cryptosystem is a cryptographic protocol that has two different keys. One of them is a public-key that can be known by everyone, and the other is the private-key only known to the user of the system. The data encrypted with the public-key of a given user can only be decrypted by this user with his or her private-key. In this paper, we propose a public-key cryptosystem based on the error-correcting codes. The decryption is performed by using the bounded distance decoding of the code. For a given code length, dimension, and error-correcting capacity, the new system allows dealing with larger plaintext than other code based public-key cryptosystems.

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26

Ni, Xiaotong. "Neural Network Decoders for Large-Distance 2D Toric Codes." Quantum 4 (August 24, 2020): 310. http://dx.doi.org/10.22331/q-2020-08-24-310.

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We still do not have perfect decoders for topological codes that can satisfy all needs of different experimental setups. Recently, a few neural network based decoders have been studied, with the motivation that they can adapt to a wide range of noise models, and can easily run on dedicated chips without a full-fledged computer. The later feature might lead to fast speed and the ability to operate at low temperatures. However, a question which has not been addressed in previous works is whether neural network decoders can handle 2D topological codes with large distances. In this work, we provide a positive answer for the toric code \cite{Kitaev2003Faulttolerantanyon}. The structure of our neural network decoder is inspired by the renormalization group decoder \cite{duclos2010fast, duclos2013fault}. With a fairly strict policy on training time, when the bit-flip error rate is lower than 9% and syndrome extraction is perfect, the neural network decoder performs better when code distance increases. With a less strict policy, we find it is not hard for the neural decoder to achieve a performance close to the minimum-weight perfect matching algorithm. The numerical simulation is done up to code distance d=64. Last but not least, we describe and analyze a few failed approaches. They guide us to the final design of our neural decoder, but also serve as a caution when we gauge the versatility of stock deep neural networks. The source code of our neural decoder can be found at \cite{sourcecodegithub}.

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27

Karner, Thomas, Brigitte Weninger, Sabine Schuster, Stefan Fleck, and Ingrid Kaminger. "Improving road freight transport statistics by using a distance matrix." Austrian Journal of Statistics 46, no. 2 (January 4, 2017): 65–80. http://dx.doi.org/10.17713/ajs.v46i2.576.

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Distances driven by road freight vehicles are an essential parameter for the calculation of transport volume. In the Austrian road freight survey, places of loading and unloading are recorded on a postal code basis. To derive the actual distances driven from this data, Statistics Austria uses a distance matrix that was first created in the 1980s. While the first version of this matrix was based on manual measurements, it has recently been recreated and updated using modern routing software. This article describes the methodology on which the current Austrian distance matrix is based. The main points discussed are: how to determine representative centroids for postal code areas; how to deal with journeys within one postal code area; and how to calculate the actual distances using routing software. The last part of the article compares the distance matrix to odometer readings from the Austrian road freight survey of the reference year 2015. This comparison showed a high positive correlation which indicates the good quality of the developed distance matrix and emphasises its usefulness in road freight transport statistics.

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28

Newland, James. "Teaching with Code: Globular Cluster Distance Lab." Research Notes of the AAS 4, no. 7 (July 28, 2020): 118. http://dx.doi.org/10.3847/2515-5172/aba953.

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29

Conway, J. H., S. J. Lomonaco, and N. J. A. Sloane. "A [45,13] code with minimal distance 16." Discrete Mathematics 83, no. 2-3 (August 1990): 213–17. http://dx.doi.org/10.1016/0012-365x(90)90007-5.

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30

HE, M. "Genetic code, hamming distance and stochastic matrices." Bulletin of Mathematical Biology 66, no. 5 (September 2004): 1405–21. http://dx.doi.org/10.1016/j.bulm.2004.01.002.

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31

Klappenecker, Andreas, and Pradeep Kiran Sarvepalli. "On subsystem codes beating the quantum Hamming or Singleton bound." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, no. 2087 (August 21, 2007): 2887–905. http://dx.doi.org/10.1098/rspa.2007.0028.

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Subsystem codes are a generalization of noiseless subsystems, decoherence-free subspaces and stabilizer codes. We generalize the quantum Singleton bound to q -linear subsystem codes. It follows that no subsystem code over a prime field can beat the quantum Singleton bound. On the other hand, we show the remarkable fact that there exist impure subsystem codes beating the quantum Hamming bound. A number of open problems concern the comparison in the performance of stabilizer and subsystem codes. One of the open problems suggested by Poulin's work asks whether a subsystem code can use fewer syndrome measurements than an optimal q -linear maximum distance separable stabilizer code while encoding the same number of qudits and having the same distance. We prove that linear subsystem codes cannot offer such an improvement under complete decoding.

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32

Latif, Asma, Kristian D. Stensland, Ryan Hendricks, Erin L. Moshier, James H. Godbold, William K. Oh, and Matt D. Galsky. "Cancer clinical trial accessibility in the United States: An analysis of travel distance." Journal of Clinical Oncology 31, no. 15_suppl (May 20, 2013): 6540. http://dx.doi.org/10.1200/jco.2013.31.15_suppl.6540.

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6540 Background: Accessibility of cancer clinical trials has been cited as a barrier to participation. While there are several dimensions to accessibility, travel distance may represent an important measure with potential socio-demographic implications. We sought to identify the driving distance from each ZIP-code in the United States to the nearest clinical trial site for four common solid tumors, and correlate ZIP-code level demographics with travel distance. Methods: The ClinicalTrials.gov database was queried on September 12, 2012 to identify all open, actively recruiting phase II and phase III therapeutic interventional trials in first-line metastatic disease for the four most common solid tumor types in the United States (prostate, breast, lung, and colorectal). Driving distance from each ZIP-code in the continental United States to the nearest trial site for each tumor type was calculated. Trial sites located within the ZIP-code were set at a travel distance of 0 miles. ZIP-code level demographics, derived from the 2010 Census, were correlated with driving distance. Results: We identified 42 prostate cancer trials with 958 sites, 81 breast cancer trials with 1,345 sites, 83 lung cancer trials with 2,249 sites, and 32 colorectal cancer trials with 566 sites which met inclusion criteria. The travel distances for each tumor type are shown in the Table. Analyses correlating driving distance with ZIP-code level demographics are ongoing. Conclusions: Substantial heterogeneity exists regarding accessibility of cancer trials in the United States as measured by driving distance. The optimal geographic distribution of trials, the burden imposed by travel, and the degree to which travel distance contributes to poor cancer clinical trial enrollment all warrant further investigation. [Table: see text]

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33

Манин, Юрий Иванович, Yurii Ivanovich Manin, Матильда Марколли, and Matilde Marcolli. "Asymptotic bounds for spherical codes." Известия Российской академии наук. Серия математическая 83, no. 3 (2019): 133–57. http://dx.doi.org/10.4213/im8739.

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The set of all error-correcting codes $C$ over a fixed finite alphabet $\mathbf{F}$ of cardinality $q$ determines the set of code points in the unit square $[0,1]^2$ with coordinates $(R(C), \delta (C))$:= (relative transmission rate, relative minimal distance). The central problem of the theory of such codes consists in maximising simultaneously the transmission rate of the code and the relative minimum Hamming distance between two different code words. The classical approach to this problem explored in vast literature consists in inventing explicit constructions of "good codes" and comparing new classes of codes with earlier ones. Less classical approach studies the geometry of the whole set of code points $(R,\delta)$ (with $q$ fixed), at first independently of its computability properties, and only afterwards turning to the problems of computability, analogies with statistical physics etc. The main purpose of this article consists in extending this latter strategy to the domain of spherical codes. Bibliography: 14 titles.

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34

Qureshi, Mohammad A., Ho-Ling Hwang, and Shih-Miao Chin. "Comparison of Distance Estimates for Commodity Flow Survey: Great Circle Distances Versus Network-Based Distances." Transportation Research Record: Journal of the Transportation Research Board 1804, no. 1 (January 2002): 212–16. http://dx.doi.org/10.3141/1804-28.

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A study was conducted to compare distance estimates derived from great circle distances (GCD) with distance estimates derived from a network-based model. The study used a sample of shipments from the 1993 commodity flow survey (CFS). For each shipment in the sample, the distance from the zip code of origin to the zip code of destination was calculated by using the Oak Ridge National Laboratories National Highway Network and assuming that the minimum impedance path was utilized. For each of these origin–destination pairs, the GCD and several variations of the GCD also were estimated. Finally, the network-based estimates and the GCD-based estimates were statistically compared. As expected, distance estimates based on GCD were found to be different from network-based estimates. However, applying a constant circuity factor of 1.22 or using variable circuity factors based on distance category did not result in a statistical bias in these distance estimates. Examination of distance estimates at the level of origin–destination pair revealed that distance estimates could vary as much as 75%. A comparison of published values for the 1997 CFS with values derived from GCD-based distance estimates shows that approximately 5% to 35% of the GCD-based values for the 1997 CFS would fall outside a two standard error interval. Although GCD-based estimates, under some conditions, may produce unbiased estimates of the mean distance, this does not eliminate the need for network-based estimates.

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Cao, H. T., R. L. Dougherty, and H. Janwa. "A (55,16,19) binary Goppa code and related codes having large minimum distance." IEEE Transactions on Information Theory 37, no. 5 (1991): 1432–33. http://dx.doi.org/10.1109/18.133263.

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Wang, Jing, Keqin Shen, Xiangyang Liu, and Chunlei Yu. "Construction of Binary Locally Repairable Codes With Optimal Distance and Code Rate." IEEE Communications Letters 25, no. 7 (July 2021): 2109–13. http://dx.doi.org/10.1109/lcomm.2021.3075520.

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37

Gabidulin, E. M., N. I. Pilipchuk, and O. V. Trushina. "Bounds on the Cardinality of Subspace Codes with Non-maximum Code Distance." Problems of Information Transmission 57, no. 3 (July 2021): 241–47. http://dx.doi.org/10.1134/s0032946021030030.

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38

Flammia, Steven T., Jeongwan Haah, Michael J. Kastoryano, and Isaac H. Kim. "Limits on the storage of quantum information in a volume of space." Quantum 1 (April 25, 2017): 4. http://dx.doi.org/10.22331/q-2017-04-25-4.

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We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubitsn, the number of encoded qubitsk, the code distanced, the accuracy parameterδthat quantifies how well the erasure channel can be reversed, and the locality parameterℓthat specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracyϵthat is exponentially small inℓ, which is the case for perturbations of local commuting projector codes, our bound readskd2D−1≤O(n(log⁡n)2DD−1)for codes onD-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceedO(ℓn(D−1)/D)ifδ≤O(ℓn−1/D). As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distancedand the sized~of a minimal region that can support all approximate logical operators satisfiesd~d1D−1≤O(nℓDD−1), where the logical operators are accurate up toO((nδ/d)1/2)in operator norm. Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types.

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39

Pandian, P. Chella. "On covering radius of codes over R = ℤ2 + uℤ2, where u2 = 0 using chinese euclidean distance." Discrete Mathematics, Algorithms and Applications 09, no. 02 (April 2017): 1750017. http://dx.doi.org/10.1142/s1793830917500173.

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In this paper, we give lower and upper bounds on the covering radius of codes over the ring [Formula: see text] where [Formula: see text] with respect to Chinese Euclidean distance and also obtain the covering radius of various Repetition codes, Simplex codes of [Formula: see text]-Type code and [Formula: see text]-Type code. We give bounds on the covering radius for MacDonald codes of both types over [Formula: see text]

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Jimmy, Jimmy, Viny Christanti Mawardi, and Agus Budi Dharmawan. "DETEKSI KEMIRIPAN SOURCE CODE DENGAN METODE FINGERPRINT BASED DISTANCE DAN LEVENSHTEIN DISTANCE." Computatio : Journal of Computer Science and Information Systems 2, no. 1 (May 22, 2018): 101. http://dx.doi.org/10.24912/computatio.v2i1.1478.

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The ease of accessing information makes easier to do plagiarism. Plagiarism not only applied to written essay, but these day copying other people’s program is also considered as plagiarism. This paper will researching an method to automatically calculating the similarity of pair of source code. The method used are fingerprint based distance and Levenshtein Distance. The result will be measured in accuracy, precision, and recall. The result is in some dataset, levenshtein distance is better than fingerprint based distance for detecting plagiarism in student’s code especially beginner in C++.

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Liu, Xiusheng, Long Yu, and Hualu Liu. "New quantum codes from Hermitian dual-containing codes." International Journal of Quantum Information 17, no. 01 (February 2019): 1950006. http://dx.doi.org/10.1142/s0219749919500060.

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Hermitian dual-containing codes play an important role in the constructing quantum codes. In this paper, we present a new criterion of Hermitian dual-containing code based on the rank of generator matrix for a linear code. Then, using the criterion, we construct a class of new quantum maximum-distance-separable (MDS) codes and some new quantum codes.

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GOHAIN, NISHA, TAZID ALI, and ADIL AKHTAR. "LATTICE STRUCTURE AND DISTANCE MATRIX OF GENETIC CODE." Journal of Biological Systems 23, no. 03 (August 30, 2015): 485–504. http://dx.doi.org/10.1142/s0218339015500254.

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The genetic code is the rule by which DNA stores the genetic information about formation of protein molecule. In this paper, a partial ordering is equipped on the genetic code and a lattice structure has been developed from it. The codon–anticodon interaction, hydrogen bond number and the chemical types of bases play an important role in the partial ordering. We have established some relations between the lattice structure of the genetic code and physico-chemical properties of amino acids. Taking into consideration the evolutionary importance of base positions in codons we have constructed a distance matrix for the amino acids. Further with a real life example we have demonstrated the relationship between frequently occurring mutations and codon distances.

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LIU, YANG, YUENA MA, YOUQIAN FENG, and RUIHU LI. "NEW QUANTUM CODES CONSTRUCTED FROM A CLASS OF IMPRIMITIVE BCH CODES." International Journal of Quantum Information 11, no. 01 (February 2013): 1350006. http://dx.doi.org/10.1142/s0219749913500068.

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By a careful analysis on cyclotomic cosets, the maximal designed distance δnew of narrow-sense imprimitive Euclidean dual containing q-ary BCH code of length [Formula: see text] is determined, where q is a prime power and l is odd. Our maximal designed distance δnew of dual containing narrow-sense BCH codes of length n improves upon the lower bound δmax for maximal designed distances of dual containing narrow-sense BCH codes given by Aly et al. [IEEE Trans. Inf. Theory53 (2007) 1183]. A series of non-narrow-sense dual containing BCH codes of length n, including the ones whose designed distances can achieve or exceed δnew, are given, and their dimensions are computed. Then new quantum BCH codes are constructed from these non-narrow-sense imprimitive BCH codes via Steane construction, and these new quantum codes are better than previous results in the literature.

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Katona, Melinda, Péter Bodnár, and László Nyúl. "Distance transform and template matching based methods for localization of barcodes and QR codes." Computer Science and Information Systems 17, no. 1 (2020): 161–79. http://dx.doi.org/10.2298/csis181011020k.

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Visual codes play an important role in automatic identification, which became an inseparable part of industrial processes. Thanks to the revolution of smartphones and telecommunication, it also becomes more and more popular in everyday life, containing embedded web addresses or other small informative texts. While barcode reading is straightforward in images having optimal parameters (focus, illumination, code orientation, and position), localization of code regions is still challenging in many scenarios. Every setup has its own characteristics, therefore many approaches are justifiable. Industrial applications are likely to have more fixed parameters like illumination, camera type and code size, and processing speed and accuracy are the most important requirements. In everyday use, like with smartphone cameras, a wide variety of code types, sizes, noise levels and blurring can be observed, but the processing speed is often not crucial, and the image acquisition process can be repeated in order for successful detection. In this paper, we address this problem with two novel methods for localization of 1D barcodes based on template matching and distance transformation, and a third method to detect QR codes. Our proposed approaches can simultaneously localize several different types of codes. We compare the effectiveness of the proposed methods with several approaches from the literature using public databases and a large set of synthetic images as a benchmark. The evaluation shows that the proposed methods are efficient, having 84.3% Jaccard accuracy, superior to other approaches. One of the presented approaches is an improvement on our previous work. Our template matching based method is computationally more complex, however, it can be adapted to specific code types providing high accuracy. The other method uses distance transformation, which is fast and gives rough regions of interests that can contain valid visual code candidates.

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Aliferis, P., D. Gottesman, and J. Preskill. "Quantum accuracy threshold for concatenated distance-3 code." Quantum Information and Computation 6, no. 2 (March 2006): 97–165. http://dx.doi.org/10.26421/qic6.2-1.

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We prove a new version of the quantum threshold theorem that applies to concatenation of a quantum code that corrects only one error, and we use this theorem to derive a rigorous lower bound on the quantum accuracy threshold $\varepsilon_0$. Our proof also applies to concatenation of higher-distance codes, and to noise models that allow faults to be correlated in space and in time. The proof uses new criteria for assessing the accuracy of fault-tolerant circuits, which are particularly conducive to the inductive analysis of recursive simulations. Our lower bound on the threshold, $\varepsilon_0 \ge 2.73\times 10^{-5}$ for an adversarial independent stochastic noise model, is derived from a computer-assisted combinatorial analysis; it is the best lower bound that has been rigorously proven so far.

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Berkman, Liubov, Olexandr Turovsky, Liudmyla Kyrpach, Oksana Varfolomeeva, Volodymyr Dmytrenko, and Oleksii Pokotylo. "Analyzing the code structures of multidimensional signals for a continuous information transmission channel." Eastern-European Journal of Enterprise Technologies 5, no. 9 (113) (October 31, 2021): 70–81. http://dx.doi.org/10.15587/1729-4061.2021.242357.

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One of the directions to improve the efficiency of modern telecommunication systems is the transition to the use of multidimensional signals for continuous channels of information transmission. As a result of studies carried out in recent years, it has been established that it is possible to ensure high quality of information transmission in continuous channels by combining demodulation and decoding operations into a single procedure that involves the construction of a code construct for a multidimensional signal. This paper considers issues related to estimating the possibility to improve the efficiency of continuous information transmission channel by changing the signal distance of the code structure. It has been established that the code structures of such types as a hierarchical code construct of signals, a hierarchical code construct of signals with Euclidean metric, a reversible code construct of signals, a reversible code construct of signals with Euclidean metric have the potential, when used, to increase the speed of information transmission along a continuous channel. With a signal distance reduced by 10 percent or larger, it could increase by two times or faster. The estimation of the effect of reducing a signal distance on the efficiency of certain types of code structures was carried out. It has been established that the hierarchical reversible code construct, compared to the hierarchical code construct, provides a win of up to two or more times in the speed of information transmission with a halved signal distance. Implementing the modulation procedure has no fundamental difficulties, on the condition that for each code of the code construct the encoding procedure is known when using binary codes. The results reported here make it possible to build an acceptably complex demodulation procedure according to the specified types of code structures

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GURITMAN, S. "Nonexistence Proofs for Five Ternary Linear Codes." Journal of Mathematics and Its Applications 1, no. 1 (July 1, 2002): 35. http://dx.doi.org/10.29244/jmap.1.1.35-40.

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<p>An [n,k, dh-code is a ternary linear code with length n, dimension k and minimum distance d. We prove that codes with parameters [110,6, 72h, [109,6,71h, [237,6,157b, [69,7,43h, and [120,9,75h do not exist.</p>

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Fernandez, Marcel, and Jorge J. Urroz. "A study of the separating property in Reed-Solomon codes by bounding the minimum distance." Designs, Codes and Cryptography 90, no. 2 (January 5, 2022): 427–42. http://dx.doi.org/10.1007/s10623-021-00988-z.

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AbstractAccording to their strength, the tracing properties of a code can be categorized as frameproof, separating, IPP and TA. It is known that, if the minimum distance of the code is larger than a certain threshold then the TA property implies the rest. Silverberg et al. ask if there is some kind of tracing capability left when the minimum distance falls below the threshold. Under different assumptions, several papers have given a negative answer to the question. In this paper, further progress is made. We establish values of the minimum distance for which Reed-Solomon codes do not posses the separating property.

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ISERSON, KENNETH V. "Telemedicine: A Proposal for an Ethical Code." Cambridge Quarterly of Healthcare Ethics 9, no. 3 (July 2000): 404–6. http://dx.doi.org/10.1017/s0963180100003133.

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Telemedicine encompasses medical practice, teaching, and research with real-time interactions over distances too great for unaided communication. It includes audio and video transmissions, either separately or combined, and can be done through mechanical (e.g., signal flags or lights) or electronic means (e.g., telecommunications). In many ways, telemedicine is a subset of medical informatics, itself a rapidly developing field. Prior definitions have been broader, including not only medical practice over distance, but also simple information transfer.

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Luna, German, Samuel Reid, Bianca De Sanctis, and Vlad Gheorghiu. "A combinatorial approach to quantum error correcting codes." Discrete Mathematics, Algorithms and Applications 06, no. 04 (October 10, 2014): 1450054. http://dx.doi.org/10.1142/s1793830914500542.

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Motivated from the theory of quantum error correcting codes, we investigate a combinatorial problem that involves a symmetric n-vertices colorable graph and a group of operations (coloring rules) on the graph: find the minimum sequence of operations that maps between two given graph colorings. We provide an explicit algorithm for computing the solution of our problem, which in turn is directly related to computing the distance (performance) of an underlying quantum error correcting code. Computing the distance of a quantum code is a highly non-trivial problem and our method may be of use in the construction of better codes.

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