K. Cinkler, K.-D. Kammeyer · IEEE Signal Processing Letters · 1999
This paper presents an a posteriori channel error correction method, based on subband synthesis. We estimate the position and amplitude of transmission errors using undownsampled subband signals, thus allowing correction of the subband signals and resynthesis of a nearly error-free signal. The method can be extended to multiple dimensions. Examples of images transmitted (compressed) with and without this method are shown. The results are very promising.
Cited by 1
Venkatesan Guruswami · Lecture notes in computer science · 2005
Cited by 216Open access
Johan Sebastian, Rosenkilde Nielsen, Supervisors Tom Høholdt et al. · 2010
Recently, Wu proposed in [24] a new approach to list decoding Reed-Solomon codes, quite different from the Guruswami-Sudan algorithm and its derivatives [1,2,10,18]. It promises lower running-time and has shown new perspectives of the codes. Sadly, the impact of this idea has hitherto been minimal; the contribution is indisputable, so the reason might be found in the idea’s sole exposition: a convoluted article, presenting and proving a long range of intricate results on a strict page limit. In this thesis, we give a complete, self-contained and novel presentation of Wu’s decoding algorithm. The key mechanics of the algorithm build on a combination of several previous decoders, and we begin with a full presentation of these and their background theory. We then derive Wu’s algorithm itself, with a clear division between the results in general algebraic theory, and the application of these on the decoding problem.
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Pankaj Das, Letminthang Haokip · Discrete Mathematics Letters · 2021
In this paper, we derive necessary and sufficient conditions for the existence of periodical burst error correcting linear codes. We also give the weight distribution of periodical burst errors and a bound on the largest attainable minimum weight by such a vector in the space of n-tuples. Furthermore, we provide probability of decoding error of a periodical burst error correcting code.
Cited by 1Open access
Jun Xie, Dong Li · 2009
Decoding of cyclic burst error correcting codes, such as Meggitt, error-trapping decoding, have generally been implemented using linear feedback shift registers(LFSR). However, such sequential decoding schemes are not suitable for modern high-speed channels that demand high-speed parallel decoding. Data storage and propagation delay are the main limitation. This paper proposes a parallel error-trapping decoding method for cyclic burst error correcting codes. In this method, a binary matrix T defines the entire decoding process. The decoding method can be implemented using mainly combination logic which could decrease the decoding delay.
Cited by 2
西森 秀稔, Hidetoshi Nishimori · Tokyo Tech Research Repository (Tokyo Institute of Technology) · 2019
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Hidetoshi Nishimori · Journal of the Physical Society of Japan · 1993
The conjecture of Ruján on error-correcting codes is proven. Errors in decoding of signals transmitted through noisy channels assume the smallest values when signals are decoded at a particular finite temperature. This finite-temperature decoding is compared with the conventional maximum likelihood decoding which corresponds to the T =0 case. The method of gauge transformation in the spin glass theory is useful in this argument.
Cited by 62
Luca G. Tallini, Paul Cull · UniTERAMO Research Catalog (University of Teramo) · 2001
Error-correcting codes are used in a variety of areas from computers to communications. Ideally, one simply looks at a received message which may contain errors, and decodes it into the error-free message. Unfortunately, this decoding process can be quite complicated and might not exploit the maximum error correction capabilities of the code. While for the simple Hamming code the decoding is direct, for more complicated codes decoding may be an NP-complete problem. Further optimal decoding depends on the statistical properties of the error source, which might be very complex. For these reasons we propose the use of neural networks as decoders. In this paper, we report some experimental results which show that neural networks can be used to practically solve the general decoding problem.
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Sérgio Escalera, Oriol Pujol, Petia Radeva · Lecture notes in computer science · 2006
Cited by 19Open access
Maiara Aparecida Coimbra Valentim · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) · 2019
Nesta dissertação estudamos principalmente métodos de decodificação de códigos corretores de erros lineares por meio de redes neurais apresentando algumas formas de relacionar tais conceitos. A partir do estudo de alguns tópicos da Teoria de Redes Neurais e da Teoria dos Grafos, fizemos uma comparação entre a decodificação por síndrome da Teoria Clássica dos Códigos Corretores de Erros e um algoritmo de decodificação utilizando redes neurais. Um dos principais resultados, envolvendo o modelo de Hopfield, prova que a decodificação de máxima verossimilhança em um código linear é equivalente a encontrar o máximo global de uma função de energia em uma rede neural. Os códigos lineares podem ser representados como estados estáveis das funções de energia. Assim, dado um código linear, uma rede neural pode ser construída de tal forma que cada máximo local na função de energia corresponda a uma palavra do código, e reciprocamente, cada palavra do código corresponda a um máximo local de uma função de energia.
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P. Cull, TALLINI, LUCA · Virtual Community of Pathological Anatomy (University of Castilla La Mancha) · 2001
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Marti i Olius, A. (Antonio) de, Crespo-Bofil, P. (Pedro), Etxezarreta-Martínez, J. (Josu) · Deposito Adademico Digital Universidad De Navarra (University of Navarra) · 2024
Quantum computers would prove a ground-breaking effect in several re- \nsearch fields to the advantage of our society due to their proved capacity \nfor solving some problems deemed as too complex for classical comput- \ners. Consequently, there is a generalized academic effort for constructing a \nquantum computer. Nonetheless, a real quantum computer does not suffice \nfor implementing quantum algorithms reliably, it must be fault-tolerant. \nQuantum computers undergo noise due to a phenomenon generally named \nquantum decoherence, a fault-tolerant quantum computer has the capabil- \nity to suppress the effects of decoherence to an extent. For that to happen, \na quantum computer should consider quantum error correction, a process \nin which decoherence is studied and attempted to be corrected. Within the \ncontext of quantum error correction, the information from quantum proces- \nsors is to be stored in a larger, more redundant system named a quantum \nerror correcting code. Afterwards, one can obtain a vector named syndrome \nwhich provides partial information on the effect decoherence has had on a \ncode. The process of recovering the decoherence or error that the code \nhas undergone is named decoding. This thesis studies decoders for quan- \ntum error correcting codes, their performance, complexity and adaption to \ndifferent types of quantum decoherence.
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Luca G. Tallini, Paul Cull · 2002
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Antonio de Mari i Olius · 2024
Quantum computers would prove a ground-breaking effect in several re- search fields to the advantage of our society due to their proved capacity for solving some problems deemed as too complex for classical comput- ers. Consequently, there is a generalized academic effort for constructing a quantum computer. Nonetheless, a real quantum computer does not suffice for implementing quantum algorithms reliably, it must be fault-tolerant. Quantum computers undergo noise due to a phenomenon generally named quantum decoherence, a fault-tolerant quantum computer has the capabil- ity to suppress the effects of decoherence to an extent. For that to happen, a quantum computer should consider quantum error correction, a process in which decoherence is studied and attempted to be corrected. Within the context of quantum error correction, the information from quantum proces- sors is to be stored in a larger, more redundant system named a quantum error correcting code. Afterwards, one can obtain a vector named syndrome which provides partial information on the effect decoherence has had on a code. The process of recovering the decoherence or error that the code has undergone is named decoding. This thesis studies decoders for quan- tum error correcting codes, their performance, complexity and adaption to different types of quantum decoherence.
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2008
Abstract. Error correcting output codes (ECOC) represent a successful extension of binary classifiers to address the multiclass problem. Lately, the ECOC framework was extended from the binary to the ternary case to allow classes to be ignored by a certain classifier, allowing in this way to increase the number of possible dichotomies to be selected. Nevertheless, the effect of the zero symbol by which dichotomies exclude certain classes from consideration has not been previously enough considered in the definition of the decoding strategies. In this paper, we show that by a special treatment procedure of zeros, and adjusting the weights at the rest of coded positions, the accuracy of the system can be increased. Besides, we extend the main state-of-art decoding strategies from the binary to the ternary case, and we propose two novel approaches: Laplacian and Pessimistic Beta Density Probability approaches. Tests on UCI database repository (with different sparse matrices containing different percentages of zero symbol) show that the ternary decoding techniques proposed outperform the standard decoding strategies. 1
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Ganesan Umanesan, Eiji Fujiwara · IEEE Transactions on Computers · 2004
Burst error correcting codes, such as fire codes, have traditionally been decoded using linear feedback shift registers (LFSR). However, such sequential decoding schemes are not suitable for modern ultra high-speed channels that demand high-speed parallel decoding employing only combinational logic circuitry. This work proposes a parallel decoding method for cyclic burst error correcting codes. Under this method, a binary companion matrix T defines the entire decoding process. Hence, the decoding method can be implemented using only combinational logic.
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Hanyan Cao, S. J. Zhao, Dongyang Feng et al. · Physical Review Letters · 2025
We study the exact maximum-likelihood decoding of quantum error-correcting codes. Our primary focus is on the repetition code under circuit-level noise, which serves as a fundamental basis for quantum error correction experiments and is the only code that has achieved large distances and extremely low error rates. We uncover that the repetition code under circuit-level noise has an exact solution and propose an optimal maximum-likelihood decoding algorithm called "planar." This algorithm is based on the exact solution of the spin-glass partition function on planar graphs and has polynomial computational complexity. Through extensive numerical experiments, we demonstrate that our algorithm uncovers the exact threshold for depolarizing noise and realistic superconductor SI1000 noise. Furthermore, we apply our method to analyze data from recent quantum memory experiments conducted by Google Quantum AI, revealing that part of the error floor was attributed to the decoding algorithm used by Google. Finally, we implemented the repetition code quantum memory on superconducting systems using a 72-qubit quantum chip lacking reset gates, demonstrating that even with an unknown error model, the proposed algorithm achieves a significantly lower logical error rate than the matching-based algorithm. We also discuss the application of our approach to quantum codes whose maximum-likelihood decoding problem corresponds to a spin-glass problem, including the surface code under independent code-capacity noise.
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M. Blaum, H.C.A. Tilborg, van · Data Archiving and Networked Services (DANS) · 1994
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Lmgm Ludo Tolhuizen · Data Archiving and Networked Services (DANS) · 1996
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Stanislav Bulygin, G.R. Pellikaan · 2007
The decoding of arbitrary linear block codes is accomplished by solving a system of quadratic equations by means of Buchberger’s algorithm for finding a Gröbner basis. This generalizes the algorithm of Berlekamp-Massey for decoding Reed-Solomon, Goppa and cyclic codes up to half the true minimum distance by intro-ducing the unknown syndromes as variables. The complexity of this algorithm is exponential and the complexity coefficient is measured under the assumption that the over-determined system of quadratic equations is semi-regular using the results of Bardet et al. [5]. The complexity is compared to existing bounded distance decoding algorithms. Our method can be extended to complete and generic decoding, and to finding the minimum distance and the complete weight distribution. 1
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