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Journal articles on the topic 'Error-correcting codes (Informations theory)'

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Author: Grafiati

Published: 4 June 2021

Last updated: 15 February 2022

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1

Conway, J., and N. Sloane. "Lexicographic codes: Error-correcting codes from game theory." IEEE Transactions on Information Theory 32, no. 3 (May 1986): 337–48. http://dx.doi.org/10.1109/tit.1986.1057187.

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2

Curto, Carina, Vladimir Itskov, Katherine Morrison, Zachary Roth, and Judy L. Walker. "Combinatorial Neural Codes from a Mathematical Coding Theory Perspective." Neural Computation 25, no. 7 (July 2013): 1891–925. http://dx.doi.org/10.1162/neco_a_00459.

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Shannon's seminal 1948 work gave rise to two distinct areas of research: information theory and mathematical coding theory. While information theory has had a strong influence on theoretical neuroscience, ideas from mathematical coding theory have received considerably less attention. Here we take a new look at combinatorial neural codes from a mathematical coding theory perspective, examining the error correction capabilities of familiar receptive field codes (RF codes). We find, perhaps surprisingly, that the high levels of redundancy present in these codes do not support accurate error correction, although the error-correcting performance of receptive field codes catches up to that of random comparison codes when a small tolerance to error is introduced. However, receptive field codes are good at reflecting distances between represented stimuli, while the random comparison codes are not. We suggest that a compromise in error-correcting capability may be a necessary price to pay for a neural code whose structure serves not only error correction, but must also reflect relationships between stimuli.

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3

Huang, Pengfei, Yi Liu, Xiaojie Zhang, Paul H. Siegel, and Erich F. Haratsch. "Syndrome-Coupled Rate-Compatible Error-Correcting Codes: Theory and Application." IEEE Transactions on Information Theory 66, no. 4 (April 2020): 2311–30. http://dx.doi.org/10.1109/tit.2020.2966439.

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4

Namba, Kazuteru, and Eiji Fujiwara. "Nonbinary single-symbol error correcting, adjacent two-symbol transposition error correcting codes over integer rings." Systems and Computers in Japan 38, no. 8 (2007): 54–60. http://dx.doi.org/10.1002/scj.10516.

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5

Ben-Gal, Irad, and Lev B. Levitin. "An application of information theory and error-correcting codes to fractional factorial experiments." Journal of Statistical Planning and Inference 92, no. 1-2 (January 2001): 267–82. http://dx.doi.org/10.1016/s0378-3758(00)00165-8.

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Namba, Kazuteru, and Eiji Fujiwara. "A class of systematicm-ary single-symbol error correcting codes." Systems and Computers in Japan 32, no. 6 (2001): 21–28. http://dx.doi.org/10.1002/scj.1030.

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7

Shimada, Ryosaku, Ryutaro Murakami, Kazuharu Sono, and Yoshiteru Ohkura. "Arithmetic burst error correcting fire-type cyclic ST-AN codes." Systems and Computers in Japan 18, no. 7 (1987): 57–68. http://dx.doi.org/10.1002/scj.4690180706.

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8

He, Xianmang. "Constructing new q-ary quantum MDS codes with distances bigger than q/2 from generator matrices." Quantum Information and Computation 18, no. 3&4 (March 2018): 223–30. http://dx.doi.org/10.26421/qic18.3-4-3.

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The construction of quantum error-correcting codes has been an active field of quantum information theory since the publication of \cite{Shor1995Scheme,Steane1998Enlargement,Laflamme1996Perfect}. It is becoming more and more difficult to construct some new quantum MDS codes with large minimum distance. In this paper, based on the approach developed in the paper \cite{NewHeMDS2016}, we construct several new classes of quantum MDS codes. The quantum MDS codes exhibited here have not been constructed before and the distance parameters are bigger than q/2.

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9

Kuznetsov, Alexandr, Oleg Oleshko, and Kateryna Kuznetsova. "ENERGY GAIN FROM ERROR-CORRECTING CODING IN CHANNELS WITH GROUPING ERRORS." Acta Polytechnica 60, no. 1 (March 2, 2020): 65–72. http://dx.doi.org/10.14311/ap.2020.60.0065.

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Abstract. This article explores the a mathematical model of the a data transmission channel with errors grouping. We propose an estimating method for energy gain from coding and energy efficiency of binary codes in channels with grouped errors. The proposed method uses a simplified Bennet and Froelich’s model and allows leading the research of the energy gain from coding for a wide class of data channels without restricting the way of the length distributing the error bursts. The reliability of the obtained results is confirmed by the information of the known results in the theory of error-correcting coding in the simplified variant.

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10

Haselgrove, H. L., and P. P. Rohde. "Trade-off between the tolerance of located and unlocated errors in nondegenrate quantum." Quantum Information and Computation 8, no. 5 (May 2008): 399–410. http://dx.doi.org/10.26421/qic8.5-3.

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In a recent study [Rohde et al., quant-ph/0603130 (2006)] of several quantum error correcting protocols designed for tolerance against qubit loss, it was shown that these protocols have the undesirable effect of magnifying the effects of depolarization noise. This raises the question of which general properties of quantum error-correcting codes might explain such an apparent trade-off between tolerance to located and unlocated error types. We extend the counting argument behind the well-known quantum Hamming bound to derive a bound on the weights of combinations of located and unlocated errors which are correctable by nondegenerate quantum codes. Numerical results show that the bound gives an excellent prediction to which combinations of unlocated and located errors can be corrected {\em with high probability} by certain large degenerate codes. The numerical results are explained partly by showing that the generalized bound, like the original, is closely connected to the information-theoretic quantity the {\em quantum coherent information}. However, we also show that as a measure of the exact performance of quantum codes, our generalized Hamming bound is provably far from tight.

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11

Mundici, Daniele. "Ulam Games, Łukasiewicz Logic, and AF C*-Algebras." Fundamenta Informaticae 18, no. 2-4 (April 1, 1993): 151–61. http://dx.doi.org/10.3233/fi-1993-182-405.

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Ulam asked what is the minimum number of yes-no questions necessary to find an unknown number in the search space (1, …, 2n), if up to l of the answers may be erroneous. The solutions to this problem provide optimal adaptive l error correcting codes. Traditional, nonadaptive l error correcting codes correspond to the particular case when all questions are formulated before all answers. We show that answers in Ulam’s game obey the (l+2)-valued logic of Łukasiewicz. Since approximately finite-dimensional (AF) C*-algebras can be interpreted in the infinite-valued sentential calculus, we discuss the relationship between game-theoretic notions and their C*-algebraic counterparts. We describe the correspondence between continuous trace AF C*-algebras, and Ulam games with separable Boolean search space S. whose questions are the clopen subspaces of S. We also show that these games correspond to finite products of countable Post MV algebras, as well as to countable lattice-ordered Specker groups with strong unit.

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12

Magdalena de la Fuente, Julio Carlos, Nicolas Tarantino, and Jens Eisert. "Non-Pauli topological stabilizer codes from twisted quantum doubles." Quantum 5 (February 17, 2021): 398. http://dx.doi.org/10.22331/q-2021-02-17-398.

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It has long been known that long-ranged entangled topological phases can be exploited to protect quantum information against unwanted local errors. Indeed, conditions for intrinsic topological order are reminiscent of criteria for faithful quantum error correction. At the same time, the promise of using general topological orders for practical error correction remains largely unfulfilled to date. In this work, we significantly contribute to establishing such a connection by showing that Abelian twisted quantum double models can be used for quantum error correction. By exploiting the group cohomological data sitting at the heart of these lattice models, we transmute the terms of these Hamiltonians into full-rank, pairwise commuting operators, defining commuting stabilizers. The resulting codes are defined by non-Pauli commuting stabilizers, with local systems that can either be qubits or higher dimensional quantum systems. Thus, this work establishes a new connection between condensed matter physics and quantum information theory, and constructs tools to systematically devise new topological quantum error correcting codes beyond toric or surface code models.

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13

Maltiyar, Kaveri, and Deepti Malviya. "Polar Code: An Advanced Encoding And Decoding Architecture For Next Generation 5G Applications." International Journal on Recent and Innovation Trends in Computing and Communication 7, no. 5 (June 4, 2019): 26–29. http://dx.doi.org/10.17762/ijritcc.v7i5.5307.

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Polar Codes become a new channel coding, which will be common to apply for next-generation wireless communication systems. Polar codes, introduced by Arikan, achieves the capacity of symmetric channels with “low encoding and decoding complexity” for a large class of underlying channels. Recently, polar code has become the most favorable error correcting code in the viewpoint of information theory due to its property of channel achieving capacity. Polar code achieves the capacity of the class of symmetric binary memory less channels. In this paper review of polar code, an advanced encoding and decoding architecture for next generation applications.

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Imai, H., J. Mueller-Quade, A. C. A. Nascimento, P. Tuyls, and A. Winter. "An information theoretical model for quantum secret sharing schemes." Quantum Information and Computation 5, no. 1 (January 2005): 68–79. http://dx.doi.org/10.26421/qic5.1-7.

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Similarly to earlier models for quantum error correcting codes, we introduce a quantum information theoretical model for quantum secret sharing schemes. This model provides new insights into the theory of quantum secret sharing. By using our model, among other results, we give a shorter proof of Gottesman's theorem that the size of the shares in a quantum secret sharing scheme must be as large as the secret itself. Also, we introduced approximate quantum secret sharing schemes and showed robustness of quantum secret sharing schemes by extending Gottesman's theorem to the approximate case.

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15

Semerenko, Vasyl, and Oleksandr Voinalovich. "The simplification of computationals in error correction coding." Technology audit and production reserves 3, no. 2(59) (June 30, 2021): 24–28. http://dx.doi.org/10.15587/2706-5448.2021.233656.

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The object of research is the processes of error correction transformation of information in automated systems. The research is aimed at reducing the complexity of decoding cyclic codes by combining modern mathematical models and practical tools. The main prerequisite for the complication of computations in deterministic linear error-correcting codes is the use of the algebraic representation as the main mathematical apparatus for these types of codes. Despite the universalism of the algebraic approach, its main drawback is the impossibility of taking into account the characteristic features of all subclasses of linear codes. In particular, the cyclic property is not taken into account at all for cyclic codes. Taking this property into account, one can go to a fundamentally different mathematical representation of cyclic codes – the theory of linear automata in Galois fields (linear finite-state machine). For the automaton representation of cyclic codes, it is proved that the problem of syndromic decoding of these codes in the general case is an NP-complete problem. However, if to use the proposed hierarchical approach to problems of complexity, then on its basis it is possible to carry out a more accurate analysis of the growth of computational complexity. Correction of single errors during one time interval (one iteration) of decoding has a linear decoding complexity on the length of the codeword, and error correction during m iterations of permutations of codeword bits has a polynomial complexity. According to three subclasses of cyclic codes, depending on the complexity of their decoding: easy decoding (linear complexity), iteratively decoded (polynomial complexity), complicate decoding (exponential complexity). Practical ways to reduce the complexity of computations are considered: alternate use of probabilistic and deterministic linear codes, simplification of software and hardware implementation by increasing the decoding time, use of interleaving. A method of interleaving is proposed, which makes it possible to simultaneously generate the burst errors and replace them with single errors. The mathematical apparatus of linear automata allows solving together the indicated problems of error correction coding.

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16

Weaver, Nik. "Quantum Graphs as Quantum Relations." Journal of Geometric Analysis 31, no. 9 (January 13, 2021): 9090–112. http://dx.doi.org/10.1007/s12220-020-00578-w.

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AbstractThe “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We use this perspective to interpret the Knill–Laflamme error-correction conditions (Knill and Laflamme in Theory of quantum error-correcting codes. Phys Rev A 55:900-911, 1997) in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke’s noncommutative graph homomorphisms (Stahlke in Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans Inf Theory 62:554–577, 2016) and Duan, Severini, and Winter’s noncommutative bipartite graphs (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013), and to realize the noncommutative confusability graph associated to a quantum channel (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013) as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the “mixed” setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.

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17

Klimo, Martin, Peter Lukáč, and Peter Tarábek. "Deep Neural Networks Classification via Binary Error-Detecting Output Codes." Applied Sciences 11, no. 8 (April 15, 2021): 3563. http://dx.doi.org/10.3390/app11083563.

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One-hot encoding is the prevalent method used in neural networks to represent multi-class categorical data. Its success stems from its ease of use and interpretability as a probability distribution when accompanied by a softmax activation function. However, one-hot encoding leads to very high dimensional vector representations when the categorical data’s cardinality is high. The Hamming distance in one-hot encoding is equal to two from the coding theory perspective, which does not allow detection or error-correcting capabilities. Binary coding provides more possibilities for encoding categorical data into the output codes, which mitigates the limitations of the one-hot encoding mentioned above. We propose a novel method based on Zadeh fuzzy logic to train binary output codes holistically. We study linear block codes for their possibility of separating class information from the checksum part of the codeword, showing their ability not only to detect recognition errors by calculating non-zero syndrome, but also to evaluate the truth-value of the decision. Experimental results show that the proposed approach achieves similar results as one-hot encoding with a softmax function in terms of accuracy, reliability, and out-of-distribution performance. It suggests a good foundation for future applications, mainly classification tasks with a high number of classes.

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18

Riznyk, V. V., D. Yu Skrybaylo-Leskiv, V. M. Badz, C. I. Hlod, V. V. Liakh, Y. M. Kulyk, N. B. Romanjuk, K. I. Tkachuk, and V. V. Ukrajinets. "COMPARATIVE ANALYSIS OF MONOLITHIC AND CYCLIC NOISE-PROTECTIVE CODES EFFECTIVENESS." Ukrainian Journal of Information Technology 3, no. 1 (2021): 99–105. http://dx.doi.org/10.23939/ujit2021.03.099.

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Comparative analysis of the effectiveness of monolithic and cyclic noise protective codes built on "Ideal Ring Bundles" (IRBs) as the common theoretical basis for synthesis, researches and application of the codes for improving technical indexes of coding systems with respect to performance, reliability, transformation speed, and security has been realized. IRBs are cyclic sequences of positive integers, which form perfect partitions of a finite interval of integers. Sums of connected IRB elements enumerate the natural integers set exactly R-times. The IRB-codes both monolithic and cyclic ones forming on the underlying combinatorial constructions can be used for finding optimal solutions for configure of an applicable coding systems based on the common mathematical platform. The mathematical model of noise-protective data coding systems presents remarkable properties of harmonious developing real space. These properties allow configure codes with useful possibilities. First of them belong to the self-correcting codes due to monolithic arranged both symbols "1" and of course "0" of each allowed codeword. This allows you to automatically detect and correct errors by the monolithic structure of the encoded words. IRB codes of the second type provide improving noise protection of the codes by choosing the optimal ratio of information parameters. As a result of comparative analysis of cyclic IRB-codes based with optimized parameters and monolithic IRB-codes, it was found that optimized cyclic IRB codes have an advantage over monolithic in relation to a clearly fixed number of detected and corrected codes, while monolithic codes favorably differ in the speed of message decoding due to their inherent properties of self-correction and encryption. Monolithic code characterized by packing of the same name characters in the form of solid blocks. The latter are capable of encoding data on several levels at the same time, which expands the ability to encrypt and protect encoded data from unauthorized access. Evaluation of the effectiveness of coding optimization methods by speed of formation of coding systems, method power, and error correcting has been made. The model based on the combinatorial configurations contemporary theory, which can find a wide scientific field for the development of fundamental and applied researches into information technolodies, including application multidimensional models, as well as algorithms for synthesis of the underlying models.

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Kesselring, Markus S., Fernando Pastawski, Jens Eisert, and Benjamin J. Brown. "The boundaries and twist defects of the color code and their applications to topological quantum computation." Quantum 2 (October 19, 2018): 101. http://dx.doi.org/10.22331/q-2018-10-19-101.

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The color code is both an interesting example of an exactly solved topologically ordered phase of matter and also among the most promising candidate models to realize fault-tolerant quantum computation with minimal resource overhead. The contributions of this work are threefold. First of all, we build upon the abstract theory of boundaries and domain walls of topological phases of matter to comprehensively catalog the objects realizable in color codes. Together with our classification we also provide lattice representations of these objects which include three new types of boundaries as well as a generating set for all 72 color code twist defects. Our work thus provides an explicit toy model that will help to better understand the abstract theory of domain walls. Secondly, we discover a number of interesting new applications of the cataloged objects for quantum information protocols. These include improved methods for performing quantum computations by code deformation, a new four-qubit error-detecting code, as well as families of new quantum error-correcting codes we call stellated color codes, which encode logical qubits at the same distance as the next best color code, but using approximately half the number of physical qubits. To the best of our knowledge, our new topological codes have the highest encoding rate of local stabilizer codes with bounded-weight stabilizers in two dimensions. Finally, we show how the boundaries and twist defects of the color code are represented by multiple copies of other phases. Indeed, in addition to the well studied comparison between the color code and two copies of the surface code, we also compare the color code to two copies of the three-fermion model. In particular, we find that this analogy offers a very clear lens through which we can view the symmetries of the color code which gives rise to its multitude of domain walls.

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20

Yu, Mian Shui, Yu Xie, and Xiao Meng Xie. "Age Classification Based on Feature Fusion." Applied Mechanics and Materials 519-520 (February 2014): 644–50. http://dx.doi.org/10.4028/www.scientific.net/amm.519-520.644.

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Age classification based on facial images is attracting wide attention with its broad application to human-computer interaction (HCI). Since human senescence is a tremendously complex process, age classification is still a highly challenging issue. In our study, Local Directional Pattern (LDP) and Gabor wavelet transform were used to extract global and local facial features, respectively, that were fused based on information fusion theory. The Principal Component Analysis (PCA) method was used for dimensionality reduction of the fused features, to obtain a lower-dimensional age characteristic vector. A Support Vector Machine (SVM) multi-class classifier with Error Correcting Output Codes (ECOC) was proposed in the paper. This was aimed at multi-class classification problems, such as age classification. Experiments on a public FG-NET age database proved the efficiency of our method.

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21

ISOKAWA, TEIJIRO, FUKUTARO ABO, FERDINAND PEPER, SUSUMU ADACHI, JIA LEE, NOBUYUKI MATSUI, and SHINRO MASHIKO. "FAULT-TOLERANT NANOCOMPUTERS BASED ON ASYNCHRONOUS CELLULAR AUTOMATA." International Journal of Modern Physics C 15, no. 06 (July 2004): 893–915. http://dx.doi.org/10.1142/s0129183104006327.

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Cellular Automata (CA) are a promising architecture for computers with nanometer-scale sized components, because their regular structure potentially allows chemical manufacturing techniques based on self-organization. With the increase in integration density, however, comes a decrease in the reliability of the components from which such computers will be built. This paper employs BCH error-correcting codes to construct CA with improved reliability. We construct an asynchronous CA of which a quarter of the (ternary) bits storing a cell's state information may be corrupted without affecting the CA's operations, provided errors are evenly distributed over a cell's bits (no burst errors allowed). Under the same condition, the corruption of half of a cell's bits can be detected.

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22

Taubin, Feliks, and Andrey Trofimov. "Concatenated Coding for Multilevel Flash Memory with Low Error Correction Capabilities in Outer Stage." SPIIRAS Proceedings 18, no. 5 (September 19, 2019): 1149–81. http://dx.doi.org/10.15622/sp.2019.18.5.1149-1181.

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One of the approaches to organization of error correcting coding for multilevel flash memory is based on concatenated construction, in particular, on multidimensional lattices for inner coding. A characteristic feature of such structures is the dominance of the complexity of the outer decoder in the total decoder complexity. Therefore the concatenated construction with low-complexity outer decoder may be attractive since in practical applications the decoder complexity is the crucial limitation for the usage of the error correction coding. We consider a concatenated coding scheme for multilevel flash memory with the Barnes-Wall lattice based codes as an inner code and the Reed-Solomon code with correction up to 4…5 errors as an outer one. Performance analysis is fulfilled for a model characterizing the basic physical features of a flash memory cell with non-uniform target voltage levels and noise variance dependent on the recorded value (input-dependent additive Gaussian noise, ID-AGN). For this model we develop a modification of our approach for evaluation the error probability for the inner code. This modification uses the parallel structure of the inner code trellis which significantly reduces the computational complexity of the performance estimation. We present numerical examples of achievable recording density for the Reed-Solomon codes with correction up to four errors as the outer code for wide range of the retention time and number of write/read cycles.

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23

Paler, Alexandru, Austin G. Fowler, and Robert Wille. "Online scheduled execution of quantum circuits protected by surface codes." Quantum Information and Computation 17, no. 15&16 (December 2017): 1335–48. http://dx.doi.org/10.26421/qic17.15-16-5.

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Quantum circuits are the preferred formalism for expressing quantum information processing tasks. Quantum circuit design automation methods mostly use a waterfall approach and consider that high level circuit descriptions are hardware agnostic. This assumption has lead to a static circuit perspective: the number of quantum bits and quantum gates is determined before circuit execution and everything is considered reliable with zero probability of failure. Many different schemes for achieving reliable fault-tolerant quantum computation exist, with different schemes suitable for different architectures. A number of large experimental groups are developing architectures well suited to being protected by surface quantum error correcting codes. Such circuits could include unreliable logical elements, such as state distillation, whose failure can be determined only after their actual execution. Therefore, practical logical circuits, as envisaged by many groups, are likely to have a dynamic structure. This requires an online scheduling of their execution: one knows for sure what needs to be executed only after previous elements have finished executing. This work shows that scheduling shares similarities with place and route methods. The work also introduces the first online schedulers of quantum circuits protected by surface codes. The work also highlights scheduling efficiency by comparing the new methods with state of the art static scheduling of surface code protected fault-tolerant circuits.

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Baldi, Marco, Alessandro Barenghi, Franco Chiaraluce, Gerardo Pelosi, and Paolo Santini. "A Finite Regime Analysis of Information Set Decoding Algorithms." Algorithms 12, no. 10 (October 1, 2019): 209. http://dx.doi.org/10.3390/a12100209.

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Decoding of random linear block codes has been long exploited as a computationally hard problem on which it is possible to build secure asymmetric cryptosystems. In particular, both correcting an error-affected codeword, and deriving the error vector corresponding to a given syndrome were proven to be equally difficult tasks. Since the pioneering work of Eugene Prange in the early 1960s, a significant research effort has been put into finding more efficient methods to solve the random code decoding problem through a family of algorithms known as information set decoding. The obtained improvements effectively reduce the overall complexity, which was shown to decrease asymptotically at each optimization, while remaining substantially exponential in the number of errors to be either found or corrected. In this work, we provide a comprehensive survey of the information set decoding techniques, providing finite regime temporal and spatial complexities for them. We exploit these formulas to assess the effectiveness of the asymptotic speedups obtained by the improved information set decoding techniques when working with code parameters relevant for cryptographic purposes. We also delineate computational complexities taking into account the achievable speedup via quantum computers and similarly assess such speedups in the finite regime. To provide practical grounding to the choice of cryptographically relevant parameters, we employ as our validation suite the ones chosen by cryptosystems admitted to the second round of the ongoing standardization initiative promoted by the US National Institute of Standards and Technology.

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Lee, Ming-Che, Jia-Wei Chang, Tzone I. Wang, and Zi Feng Huang. "Using Variation Theory as a Guiding Principle in an OOP Assisted Syntax Correction Learning System." International Journal of Emerging Technologies in Learning (iJET) 15, no. 14 (July 31, 2020): 35. http://dx.doi.org/10.3991/ijet.v15i14.14191.

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Object-oriented programming skill is important for the software professionals. It has become a mandatory course in information science and computer engineering departments of universities. However, it is hard for novice learners to understand the syntax and semantics of the language while learning object-oriented programming, and that makes them feel frustrated. The purpose of this study is to build an object-oriented programming assistant system that gives syntax error feedback based the variation theory. We established the syntax correction module on the basis of the Virtual Teaching Assistant (VTA). While compiling codes, the system will display syntax errors, if any, with feedbacks that are designed according to the variation theory in different levels (the generation, contrast, separation, and fusion levels) to help them correcting the errors. The experiment design of this study splits the participants, who are university freshmen, into two groups by the S-type method based on the result of a mid-term test. The learning performances and questionnaires were used for surveying, followed by in-depth inter-views, to evaluate the feasibility of the proposed assistant system. The findings indicate that the learners in the experimental group achieved better learning outcomes than their counterparts in the control group. This can also prove that the strategy of using the variation theory in implementing feed-back for object-oriented programming is effective.

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Haeupler, Bernhard, and Amirbehshad Shahrasbi. "Synchronization Strings: Codes for Insertions and Deletions Approaching the Singleton Bound." Journal of the ACM 68, no. 5 (October 31, 2021): 1–39. http://dx.doi.org/10.1145/3468265.

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We introduce synchronization strings , which provide a novel way to efficiently deal with synchronization errors , i.e., insertions and deletions. Synchronization errors are strictly more general and much harder to cope with than more commonly considered Hamming-type errors , i.e., symbol substitutions and erasures. For every ε > 0, synchronization strings allow us to index a sequence with an ε -O(1) -size alphabet, such that one can efficiently transform k synchronization errors into (1 + ε)k Hamming-type errors . This powerful new technique has many applications. In this article, we focus on designing insdel codes , i.e., error correcting block codes (ECCs) for insertion-deletion channels. While ECCs for both Hamming-type errors and synchronization errors have been intensely studied, the latter has largely resisted progress. As Mitzenmacher puts it in his 2009 survey [30]: “ Channels with synchronization errors...are simply not adequately understood by current theory. Given the near-complete knowledge, we have for channels with erasures and errors...our lack of understanding about channels with synchronization errors is truly remarkable. ” Indeed, it took until 1999 for the first insdel codes with constant rate, constant distance, and constant alphabet size to be constructed and only since 2016 are there constructions of constant rate insdel codes for asymptotically large noise rates. Even in the asymptotically large or small noise regimes, these codes are polynomially far from the optimal rate-distance tradeoff. This makes the understanding of insdel codes up to this work equivalent to what was known for regular ECCs after Forney introduced concatenated codes in his doctoral thesis 50 years ago. A straightforward application of our synchronization strings-based indexing method gives a simple black-box construction that transforms any ECC into an equally efficient insdel code with only a small increase in the alphabet size. This instantly transfers much of the highly developed understanding for regular ECCs into the realm of insdel codes. Most notably, for the complete noise spectrum, we obtain efficient “near-MDS” insdel codes, which get arbitrarily close to the optimal rate-distance tradeoff given by the Singleton bound. In particular, for any δ ∈ (0,1) and ε > 0, we give a family of insdel codes achieving a rate of 1 - δ - ε over a constant-size alphabet that efficiently corrects a δ fraction of insertions or deletions.

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Newman, Michael, and Yaoyun Shi. "Limitations on transversal computation through quantum homomorphic encryption." Quantum Information and Computation 18, no. 11&12 (September 2018): 927–48. http://dx.doi.org/10.26421/qic18.11-12-3.

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Transversality is a simple and effective method for implementing quantum computation fault-tolerantly. However, no quantum error-correcting code (QECC) can transversally implement a quantum universal gate set (Eastin and Knill, {\em Phys. Rev. Lett.}, 102, 110502). Since reversible classical computation is often a dominating part of useful quantum computation, whether or not it can be implemented transversally is an important open problem. We show that, other than a small set of non-additive codes that we cannot rule out, no binary QECC can transversally implement a classical reversible universal gate set. In particular, no such QECC can implement the Toffoli gate transversally.}{We prove our result by constructing an information theoretically secure (but inefficient) quantum homomorphic encryption (ITS-QHE) scheme inspired by Ouyang {\em et al.} (arXiv:1508.00938). Homomorphic encryption allows the implementation of certain functions directly on encrypted data, i.e. homomorphically. Our scheme builds on almost any QECC, and implements that code's transversal gate set homomorphically. We observe a restriction imposed by Nayak's bound ({\em FOCS} 1999) on ITS-QHE, implying that any ITS quantum {\em fully} homomorphic scheme (ITS-QFHE) implementing the full set of classical reversible functions must be highly inefficient. While our scheme incurs exponential overhead, any such QECC implementing Toffoli transversally would still violate this lower bound through our scheme.

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Garzon, Max H., and Kiran C. Bobba. "Geometric Approaches to Gibbs Energy Landscapes and DNA Oligonucleotide Design." International Journal of Nanotechnology and Molecular Computation 3, no. 3 (July 2011): 42–56. http://dx.doi.org/10.4018/ijnmc.2011070104.

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DNA codeword design has been a fundamental problem since the early days of DNA computing. The problem calls for finding large sets of single DNA strands that do not crosshybridize to themselves, to each other or to others' complements. Such strands represent so-called domains, particularly in the language of chemical reaction networks (CRNs). The problem has shown to be of interest in other areas as well, including DNA memories and phylogenetic analyses because of their error correction and prevention properties. In prior work, a theoretical framework to analyze this problem has been developed and natural and simple versions of Codeword Design have been shown to be NP-complete using any single reasonable metric that approximates the Gibbs energy, thus practically making it very difficult to find any general procedure for finding such maximal sets exactly and efficiently. In this framework, codeword design is partially reduced to finding large sets of strands maximally separated in DNA spaces and, therefore, the size of such sets depends on the geometry of these spaces. Here, the authors describe in detail a new general technique to embed them in Euclidean spaces in such a way that oligonucleotides with high (low, respectively) hybridization affinity are mapped to neighboring (remote, respectively) points in a geometric lattice. This embedding materializes long-held metaphors about codeword design in analogies with error-correcting code design in information theory in terms of sphere packing and leads to designs that are in some cases known to be provably nearly optimal for small oligonucleotide sizes, whenever the corresponding spherical codes in Euclidean spaces are known to be so. It also leads to upper and lower bounds on estimates of the size of optimal codes of size under 20-mers, as well as to a few infinite families of DNA strand lengths, based on estimates of the kissing (or contact) number for sphere codes in high-dimensional Euclidean spaces. Conversely, the authors show how solutions to DNA codeword design obtained by experimental or other means can also provide solutions to difficult spherical packing geometric problems via these approaches. Finally, the reduction suggests a tool to provide some insight into the approximate structure of the Gibbs energy landscapes, which play a primary role in the design and implementation of biomolecular programs.

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29

Sabin, Roberta Evans, and Samuel J. Lomonaco. "Metacyclic error-correcting codes." Applicable Algebra in Engineering, Communication and Computing 6, no. 3 (May 1995): 191–210. http://dx.doi.org/10.1007/bf01195337.

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30

Steane, A. M. "Error Correcting Codes in Quantum Theory." Physical Review Letters 77, no. 5 (July 29, 1996): 793–97. http://dx.doi.org/10.1103/physrevlett.77.793.

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31

Knill, Emanuel, and Raymond Laflamme. "Theory of quantum error-correcting codes." Physical Review A 55, no. 2 (February 1, 1997): 900–911. http://dx.doi.org/10.1103/physreva.55.900.

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32

Imai, Hideki, and Manabu Hagiwara. "Error-correcting codes and cryptography." Applicable Algebra in Engineering, Communication and Computing 19, no. 3 (April 2, 2008): 213–28. http://dx.doi.org/10.1007/s00200-008-0074-0.

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33

Smith, Stephen D., and Satoshi Yoshiara. "Group Geometries and Error-Correcting Codes." Journal of Algebra 175, no. 1 (July 1995): 199–211. http://dx.doi.org/10.1006/jabr.1995.1183.

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34

Horak, Peter. "Error-correcting codes and Minkowski’s conjecture." Tatra Mountains Mathematical Publications 45, no. 1 (December 1, 2010): 37–49. http://dx.doi.org/10.2478/v10127-010-0004-y.

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ABSTRACT The goal of this paper is twofold. The main one is to survey the latest results on the perfect and quasi-perfect Lee error correcting codes. The other goal is to show that the area of Lee error correcting codes, like many ideas in mathematics, can trace its roots to the Phytagorean theorem a2+b2 = c2. Thus to show that the area of the perfect Lee error correcting codes is an integral part of mathematics. It turns out that Minkowski’s conjecture, which is an interface of number theory, approximation theory, geometry, linear algebra, and group theory is one of the milestones on the route to Lee codes.

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35

Baylis, John, and Vera Pless. "Introduction to the Theory of Error-Correcting Codes." Mathematical Gazette 75, no. 472 (June 1991): 231. http://dx.doi.org/10.2307/3620287.

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36

Abbott, Steve, and Vera Pless. "Introduction to the Theory of Error-Correcting Codes." Mathematical Gazette 83, no. 497 (July 1999): 351. http://dx.doi.org/10.2307/3619098.

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37

N., H., and Vera Pless. "Introduction to the Theory of Error-Correcting Codes." Mathematics of Computation 56, no. 193 (January 1991): 399. http://dx.doi.org/10.2307/2008564.

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38

Gordon, Daniel M. "Perfect multiple error-correcting arithmetic codes." Mathematics of Computation 49, no. 180 (1987): 621. http://dx.doi.org/10.1090/s0025-5718-1987-0906195-7.

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39

Manin, Yuri I., and Matilde Marcolli. "Error-Correcting Codes and Phase Transitions." Mathematics in Computer Science 5, no. 2 (March 23, 2010): 133–70. http://dx.doi.org/10.1007/s11786-010-0031-8.

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40

Dahlberg, Axel, and Stephanie Wehner. "Transforming graph states using single-qubit operations." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2123 (May 28, 2018): 20170325. http://dx.doi.org/10.1098/rsta.2017.0325.

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Stabilizer states form an important class of states in quantum information, and are of central importance in quantum error correction. Here, we provide an algorithm for deciding whether one stabilizer (target) state can be obtained from another stabilizer (source) state by single-qubit Clifford operations (LC), single-qubit Pauli measurements (LPM) and classical communication (CC) between sites holding the individual qubits. What is more, we provide a recipe to obtain the sequence of LC+LPM+CC operations which prepare the desired target state from the source state, and show how these operations can be applied in parallel to reach the target state in constant time. Our algorithm has applications in quantum networks, quantum computing, and can also serve as a design tool—for example, to find transformations between quantum error correcting codes. We provide a software implementation of our algorithm that makes this tool easier to apply. A key insight leading to our algorithm is to show that the problem is equivalent to one in graph theory, which is to decide whether some graph G ′ is a vertex-minor of another graph G . The vertex-minor problem is, in general, -Complete, but can be solved efficiently on graphs which are not too complex. A measure of the complexity of a graph is the rank-width which equals the Schmidt-rank width of a subclass of stabilizer states called graph states, and thus intuitively is a measure of entanglement. Here, we show that the vertex-minor problem can be solved in time O (| G | 3 ), where | G | is the size of the graph G , whenever the rank-width of G and the size of G ′ are bounded. Our algorithm is based on techniques by Courcelle for solving fixed parameter tractable problems, where here the relevant fixed parameter is the rank width. The second half of this paper serves as an accessible but far from exhausting introduction to these concepts, that could be useful for many other problems in quantum information. This article is part of a discussion meeting issue ‘Foundations of quantum mechanics and their impact on contemporary society’.

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Achlioptas, Dimitris, and Panos Theodoropoulos. "Model counting with error-correcting codes." Constraints 24, no. 2 (February 8, 2019): 162–82. http://dx.doi.org/10.1007/s10601-018-9296-3.

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42

Nakashima, Tohru. "Error-correcting codes on projective bundles." Finite Fields and Their Applications 12, no. 2 (April 2006): 222–31. http://dx.doi.org/10.1016/j.ffa.2005.04.008.

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43

Dass, B. K., and Rashmi Verma. "REPEATED BURST ERROR CORRECTING LINEAR CODES." Asian-European Journal of Mathematics 01, no. 03 (September 2008): 303–35. http://dx.doi.org/10.1142/s1793557108000278.

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Many kinds of errors in coding theory have been dealt with for which codes have been constructed to combat such errors. Though there is a long history concerning the growth of the subject and many of the codes developed have found applications in numerous areas of practical interest, one of the areas of practical importance in which a parallel growth of the subject took place is that of burst error detecting and correcting codes. The nature of burst errors differ from channel to channel depending upon the behaviour of channels or the kind of errors which occur during the process of data transmission. In very busy communication channels, errors repeat themselves more frequently. In view of this, it is desirable to consider repeated burst errors. The paper presents lower and upper bounds on the number of parity-check digits required for a linear code correcting errors in the form of repeated bursts. An upper bound for a code that detects m-repeated bursts has also been derived. Illustrations of several codes that correct 2-repeated bursts of different lengths have also been given.

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Ballico, Edoardo, and Claudio Fontanari. "The Horace Method for Error-Correcting Codes." Applicable Algebra in Engineering, Communication and Computing 17, no. 2 (April 20, 2006): 135–39. http://dx.doi.org/10.1007/s00200-006-0012-y.

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Hossain, Md Zahangir, Partha Pratim Dey, and M. Asifuzzaman. "One-error correcting [5, 2] ternary codes." Journal of Discrete Mathematical Sciences and Cryptography 22, no. 6 (August 18, 2019): 935–41. http://dx.doi.org/10.1080/09720529.2019.1624337.

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46

Nikolos, Gaitanis, and Philokyprou. "Systematic t-Error Correcting/All Unidirectional Error Detecting Codes." IEEE Transactions on Computers C-35, no. 5 (May 1986): 394–402. http://dx.doi.org/10.1109/tc.1986.1676782.

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47

Morandi, Patrick J., and B. A. Sethuraman. "Divisors on division algebras and error correcting codes." Communications in Algebra 26, no. 10 (January 1998): 3211–21. http://dx.doi.org/10.1080/00927879808826337.

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48

Umanesan, G., and E. Fujiwara. "Parallel decoding cyclic burst error correcting codes." IEEE Transactions on Computers 54, no. 1 (January 2005): 87–92. http://dx.doi.org/10.1109/tc.2005.9.

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49

Hansen, Søren Have. "Error-Correcting Codes from Higher-Dimensional Varieties." Finite Fields and Their Applications 7, no. 4 (October 2001): 530–52. http://dx.doi.org/10.1006/ffta.2001.0313.

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50

Zarzar, Marcos. "Error-correcting codes on low rank surfaces." Finite Fields and Their Applications 13, no. 4 (November 2007): 727–37. http://dx.doi.org/10.1016/j.ffa.2007.05.001.

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