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optimal encoding of binary cyclic codes

IEICE TRANSACTIONS ON COMMUNICATIONS 2006年第12期E89B卷 3280-3287页

作者： Chen, Houshou Natl Chung Hsing Univ Dept Elect Engn Taichung 40227 Taiwan Natl Chung Hsing Univ Inst Commun Engn Taichung 40227 Taiwan

This paper considers the optimal generator matrices of a given binary cyclic code over a binary symmetric channel with crossover probability p -> 0 when the goal is to minimize the probability of an information bit error. A given code has many encoder realizations and the information bit error probability is a function of this realization. Our goal here is to seek the optimal realization of encoding functions by taking advantage of the structure of the codes, and to derive the probability of information bit error when possible. We derive some sufficient conditions for a binary cyclic code to have systematic optimal generator matrices under bounded distance decoding and determine many cyclic codes with such properties. We also present some binary cyclic codes whose optimal generator matrices are non-systematic under complete decoding.

关键词： linear block codes binary cyclic codes bit error probability optimal encoding systematic generator matrix

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A New Zigzag MDS Code with optimal encoding and Efficient Decoding 2

A New Zigzag MDS Code with Optimal Encoding and Efficient De...

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IEEE International Conference on Big Data

作者： Chen, Jun Li, Hui Hou, Hanxu Zhu, Bing Zhou, Tai Lu, Lijia Zhang, Yumeng Peking Univ Shenzhen Grad Sch Inst Big Data Technol Shenzhen Engn Lab Converged Networks Technol Shenzhen 518055 Peoples R China

ISBN: (纸本)9781479956661

Distributed file system has emerged in recent years as an efficient solution to store the large amount of data produced anytime and anywhere. In order to guarantee data reliability, it is necessary to introduce redundancy to the storage systems. Compared to simple replication, practical systems are increasingly adopting erasure codes for better storage efficiency. However, traditional erasure codes such as maximum-distanceseparable (MDS) codes, are designed over a large finite field, which inevitably hinders the wide implementation of erasure codes. In this paper, we propose a new family of MDS codes with high computation efficiency. More specifically, only XOR operation is included in the encoding process to generate parity blocks. Upon failure of a storage node, we use the efficient Zigzag decoding method to recover the failed blocks, which achieves the optimal encoding and an efficient decoding. Furthermore, we implement the proposed codes in a distributed file system, and the results show the high performance of the new codes.

关键词： Distributed file systems MDS codes optimal encoding Efficient decoding Zigzag decoding

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Efficient network coding-based multicast retransmission mechanism for mobile communication networks

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IET COMMUNICATIONS 2012年第2期6卷 187-193页

作者： Wu, H. Zheng, J. Southeast Univ Natl Mobile Commun Res Lab Nanjing 210096 Jiangsu Peoples R China Xidian Univ State Key Lab Integrated Serv Networks Xian 710071 Shaanxi Peoples R China

This study proposes a network Coding-based RETransmission (CoRET) mechanism for supporting multicast service in mobile communication networks. CoRET introduces deterministic network coding to improve retransmission efficiency and retransmission reliability. To support CoRET, a couple of efficient packet selection algorithms are proposed, one is called 'Most-Least' selection and the other is called Hamming distance-based ('Hamming-D') selection, to select packets when performing network coding for retransmission. Most-Least selects the packets with the most retransmission requests and the least retransmission requests, whereas Hamming-D selects the packets with the largest Hamming distance. Simulation results show that with Most-Least or Hamming-D CoRET outperforms an existing retransmission scheme, 'Random Pick', in terms of both retransmission efficiency and retransmission reliability. To achieve the best performance of the Hamming-D algorithm, a mathematical model is further developed to analyse the impact of the number of original packets encoded into a packet for retransmission on the retransmission performance to find an optimal encoding number that can achieve the best performance.

关键词： Reliability deterministic network coding network coding optimal encoding retransmission efficiency mobile communication Codes Mobile radio systems most-least selection least retransmission requests multicast service retransmission reliability packet selection algorithms mobile communication networks network coding-based multicast retransmission mechanism Hamming distance-based selection Hamming codes telecommunication services mathematical model telecommunication network reliability

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A CLASS OF LINEAR-MAPS FOR ERROR CORRECTIVE DIMENSIONALITY REDUCTION OF BINARY TEMPLATES

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PATTERN RECOGNITION 1988年第4期21卷 381-392页

作者： KUNDU, S INDIAN STAT INST ELECTR & COMMUN SCI UNITCALCUTTA 700035W BENGALINDIA

Binary templates are optimally encoded in a reduced dimension by a proposed class of linear maps that preserves the local neighbourhood and a prescribed minimum distance between the prototypes to a workable extent. This in effect generates a nearness criterion suitable for template matching with a level of error correcting capability in the reduced space while requiring only a fraction of memory storage space and boolean operations that would have been required otherwise. Characters and symbols may now be designed with reference to separation and shape but with a comparative freedom from the constraint of dimension, while a volume of such data can be transmitted with the speed and bandwidth of the encoded data. Some of the principal problems associated with template matching are thus overcome. Here all the necessary operations are performed in the finite field GF(2) and the methodology developed can be implemented in a microcomputer with improved system performance and economy.

关键词： Algebraic algorithm Dimensionality reduction Error correction Linear mapping Minimum distance encoding Neighbourhood preserving transformation optimal encoding Probabilistic algorithm Perfect hash function Template stylization

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optimal Program-Size Complexity for Self-Assembled Squares at Temperature 1 in 3D

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ALGORITHMICA 2017年第4期77卷 1240-1282页

作者： Furcy, David Micka, Samuel Summers, Scott M. Univ Wisconsin Oshkosh Dept Comp Sci Oshkosh WI 54901 USA Montana State Univ Dept Comp Sci Bozeman MT 59717 USA

Working in a three-dimensional variant of Winfree's abstract Tile Assembly Model, we show that, for all , there is a tile set that uniquely self-assembles into an square shape at temperature 1 with program-size complexity of (the program-size complexity, also known as tile complexity, of a shape is the minimum number of unique tile types required to uniquely self-assemble it). Note that this tile complexity is optimal for all algorithmically random values of N. Moreover, our construction is "just barely" 3D in the sense that it works even when the placement of tiles is restricted to the and planes. This result affirmatively answers an open question from Cook et al. (SODA 2011). To achieve this result, we develop a general 3D temperature 1 optimal encoding construction, reminiscent of the 2D temperature 2 optimal encoding construction of Soloveichik and Winfree (SIAM J Comput 36(6):1544-1569, 2007), and perhaps of independent interest. We also develop a more sophisticated 3D temperature 1 optimal encoding construction that places asymptotically fewer tiles than our original optimal encoding construction. We use the method of conditional determinism of Lutz and Shutters (Theory Comput Syst 51(3):372-400, 2012) to prove the correctness of our constructions.

关键词： Algorithmic self-assembly optimal encoding DNA computing Program-size complexity

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optimal Self-Assembly of Finite Shapes at Temperature 1 in 3D

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ALGORITHMICA 2018年第6期80卷 1909-1963页

作者： Furcy, David Summers, Scott M. Univ Wisconsin Dept Comp Sci Oshkosh WI 54901 USA

Tile self-assembly in which tiles may bind in a non-cooperative fashion is often referred to as "temperature 1 self-assembly" or simply "non-cooperative self-assembly". In this type of self-assembly, a tile may non-cooperatively bind to an assembly via (at least) one of its sides, unlike in cooperative self-assembly, in which some tiles may be required to bind on two or more sides. Cooperative self-assembly leads to highly non-trivial theoretical behavior but two-dimensional non-cooperative self-assembly is conjectured to be only capable of producing highly-regular shapes and patterns, which, in general, cannot be interpreted as complex computation. Remarkably, Cook et al. (Temperature 1 self-assembly: deterministic assembly in 3D and probabilistic assembly in 2D. In: Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms 2011) showed that three-dimensional non-cooperative self-assembly is essentially as powerful as cooperative self-assembly, with respect to Turing universality and self-assembly of squares. In this paper, we consider the non-cooperative self-assembly of just-barely 3D shapes. Working in a three-dimensional variant of Winfree's abstract Tile Assembly Model, we show that, for an arbitrary finite, connected shape , there is a tile set that uniquely self-assembles at temperature 1 into a 3D representation of a scaled-up version of X with optimal program-size complexity, where the "program-size complexity", also known as "tile complexity", of a shape is the minimum number of tile types required to uniquely self-assemble it. Moreover, our construction is "just barely" 3D in the sense that it only places tiles in the and planes. Our result is essentially a just-barely 3D temperature 1 simulation of a similar 2D temperature 2 result by Soloveichik and Winfree (SIAM J Comput 36(6):1544-1569, 2007).

关键词： Self-assembly Algorithmic self-assembly Kolmogorov complexity Scaled shapes optimal encoding optimal self-assembly Temperature 1 Non-cooperative self-assembly

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Bernstein polynomials and learning theory

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JOURNAL OF APPROXIMATION THEORY 2004年第2期128卷 187-206页

作者： Braess, D Sauer, T Ruhr Univ Bochum Fac Math D-44780 Bochum Germany Univ Giessen Lehrstuhl Numer Math D-35392 Giessen Germany

When learning processes depend on samples but not on the order of the information in the sample, then the Bernoulli distribution is relevant and Bernstein polynomials enter into the analysis. We derive estimates of the approximation of the entropy function x log x that are sharper than the bounds from Voronovskaja's theorem. In this way we get the correct asymptotics for the Kullback-Leibler distance for an encoding problem. (C) 2004 Elsevier Inc. All rights reserved.

关键词： Bernstein polynomials entropy function learning theory optimal encoding

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Low-Complexity Wire-Tap Codes with Security and Error-Correction Guarantees

Low-Complexity Wire-Tap Codes with Security and Error-Correc...

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IEEE Information Theory Workshop (ITW)

作者： Cassuto, Yuval Bandic, Zvonimir Ecole Polytech Fed Lausanne ALGO Sch Comp & Commun Sci CH-1015 Lausanne Switzerland San Jose Res Ctr San Jose CA 95192 USA

ISBN: (纸本)9781424482641

New code constructions are proposed for the wiretap channel with security and error-correction guarantees. For the case of error-free main channels, two families of codes are constructed with optimal encoding and decoding complexities for their wire-tap security. For the case of main channels with errors, two concatenation types are studied for the wire-tap and error-correcting codes. For each of these concatenated schemes, code families are constructed that give optimal cooperation between the wire-tap and error-correction properties. The motivation to study low-complexity wire-tap codes with security and error-correction guarantees comes from data storage applications. Due to imperfect physical erasure processes, important secret information needs to be protected from adversarial access to residual, post erasure, information, and at the same time be protected from errors when read by the legitimate device user.

关键词： channel coding code constructions code family concatenated codes concatenation types data storage applications decoding complexity error correction codes error-correcting codes error-correction guarantees error-free main channels low-complexity wire-tap codes optimal encoding physical erasure processes secret information telecommunication security wire-tap security wiretap channel

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Communication with Disturbance Constraints

Communication with Disturbance Constraints

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IEEE International Symposium on Information Theory (ISIT)

作者： Bandemer, Bernd El Gamal, Abbas Stanford Univ Informat Syst Lab Stanford CA 94305 USA

ISBN: (纸本)9781457705953

The problem of communication with disturbance constraints is introduced. The rate-disturbance region is established for the single constraint case. The optimal encoding scheme turns out to be the same as the Han-Kobayashi scheme for the two user-pair interference channel. For communication with two disturbance constraints, a coding scheme and a corresponding inner bound for the deterministic case are presented. The results suggest a natural way to obtain a new inner bound on the capacity region of the interference channel with more than two user pairs.

关键词： Decoding encoding Entropy Han-Kobayashi scheme Interference channels Noise Receivers Zinc broadcast channels capacity region disturbance constraints encoding inner bound interference (signal) optimal encoding rate disturbance region single constraint user pair interference channel

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Estimation with Communication Cost over a Finite Alphabet - a Transport Theory Approach

Estimation with Communication Cost over a Finite Alphabet - ...

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European Control Conference (ECC)

作者： Lipsa, Gabriel M. Int Monetary Fund Econ Syst Div Washington DC 20431 USA

ISBN: (纸本)9783952426937

Consider a stochastic process, a pre-processor that accepts causal measurements of the process and a state estimator. The pre-processor and the estimator are not co-located and each time the pre-processor can send a symbol to the estimator from a finite alphabet. We seek the pro-processor and the estimator that jointly minimize a cost combining two terms: the expected squared state estimation error and a communication cost, under a constraint for the communication cost. The communication cost comes from the fact that each transmission of a symbol incurs a cost. In this paper we find what is the optimal encoding at the pre-processor side and which is the optimal estimate.

关键词： state estimation stochastic processes causal measurements communication cost expected squared state estimation error finite alphabet optimal encoding preprocessor side stochastic process symbol transmission transport theory approach Couplings Estimation Geometry Measurement Optimization Random variables Stochastic processes Markov chain Stochastic Processes State estimation Program processors Random variable Couplings geometry Communication Research transport theory

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