53 research outputs found

    Raptor Codes in the Low SNR Regime

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    In this paper, we revisit the design of Raptor codes for binary input additive white Gaussian noise (BIAWGN) channels, where we are interested in very low signal to noise ratios (SNRs). A linear programming degree distribution optimization problem is defined for Raptor codes in the low SNR regime through several approximations. We also provide an exact expression for the polynomial representation of the degree distribution with infinite maximum degree in the low SNR regime, which enables us to calculate the exact value of the fractions of output nodes of small degrees. A more practical degree distribution design is also proposed for Raptor codes in the low SNR regime, where we include the rate efficiency and the decoding complexity in the optimization problem, and an upper bound on the maximum rate efficiency is derived for given design parameters. Simulation results show that the Raptor code with the designed degree distributions can approach rate efficiencies larger than 0.95 in the low SNR regime.Comment: Submitted to the IEEE Transactions on Communications. arXiv admin note: text overlap with arXiv:1510.0772

    Cooperative Local Repair in Distributed Storage

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    Erasure-correcting codes, that support local repair of codeword symbols, have attracted substantial attention recently for their application in distributed storage systems. This paper investigates a generalization of the usual locally repairable codes. In particular, this paper studies a class of codes with the following property: any small set of codeword symbols can be reconstructed (repaired) from a small number of other symbols. This is referred to as cooperative local repair. The main contribution of this paper is bounds on the trade-off of the minimum distance and the dimension of such codes, as well as explicit constructions of families of codes that enable cooperative local repair. Some other results regarding cooperative local repair are also presented, including an analysis for the well-known Hadamard/Simplex codes.Comment: Fixed some minor issues in Theorem 1, EURASIP Journal on Advances in Signal Processing, December 201

    On a Low-Rate TLDPC Code Ensemble and the Necessary Condition on the Linear Minimum Distance for Sparse-Graph Codes

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    This paper addresses the issue of design of low-rate sparse-graph codes with linear minimum distance in the blocklength. First, we define a necessary condition which needs to be satisfied when the linear minimum distance is to be ensured. The condition is formulated in terms of degree-1 and degree-2 variable nodes and of low-weight codewords of the underlying code, and it generalizies results known for turbo codes [8] and LDPC codes. Then, we present a new ensemble of low-rate codes, which itself is a subclass of TLDPC codes [4], [5], and which is designed under this necessary condition. The asymptotic analysis of the ensemble shows that its iterative threshold is situated close to the Shannon limit. In addition to the linear minimum distance property, it has a simple structure and enjoys a low decoding complexity and a fast convergence.Comment: submitted to IEEE Trans. on Communication

    Error Resilience in Distributed Storage via Rank-Metric Codes

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    This paper presents a novel coding scheme for distributed storage systems containing nodes with adversarial errors. The key challenge in such systems is the propagation of erroneous data from a single corrupted node to the rest of the system during a node repair process. This paper presents a concatenated coding scheme which is based on two types of codes: maximum rank distance (MRD) code as an outer code and optimal repair maximal distance separable (MDS) array code as an inner code. Given this, two different types of adversarial errors are considered: the first type considers an adversary that can replace the content of an affected node only once; while the second attack-type considers an adversary that can pollute data an unbounded number of times. This paper proves that the proposed coding scheme attains a suitable upper bound on resilience capacity for the first type of error. Further, the paper presents mechanisms that combine this code with subspace signatures to achieve error resilience for the second type of errors. Finally, the paper concludes by presenting a construction based on MRD codes for optimal locally repairable scalar codes that can tolerate adversarial errors

    Coding schemes for multicode CDMA systems.

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    Zhao Fei.Thesis (M.Phil.)--Chinese University of Hong Kong, 2003.Includes bibliographical references (leaves 87-89).Abstracts in English and Chinese.Chapter 1. --- Introduction --- p.1Chapter 1.1 --- Multirate Scheme --- p.2Chapter 1.1.1 --- VSF Scheme --- p.3Chapter 1.1.2 --- Multicode Scheme --- p.5Chapter 1.2 --- Multicode CDMA System --- p.7Chapter 1.2.1 --- System Model --- p.7Chapter 1.2.2 --- Envelope Variation of Multicode Signal --- p.9Chapter 1.2.3 --- Drawback of Multicode Scheme --- p.11Chapter 1.3 --- Organization of the Thesis --- p.13Chapter 2. --- Related Work on Minimization of PAP of Multicode CDMA --- p.15Chapter 2.1 --- Constant Amplitude Coding --- p.16Chapter 2.2 --- Multidimensional Multicode Scheme --- p.22Chapter 2.3 --- Precoding for Multicode Scheme --- p.25Chapter 2.4 --- Summary --- p.26Chapter 3. --- Multicode CDMA System with Constant Amplitude Transmission --- p.27Chapter 3.1 --- System Model --- p.28Chapter 3.2 --- Selection of Hadamard Code Sequences --- p.31Chapter 3.3 --- The Optimal Receiver for the Multicode System --- p.37Chapter 3.3.1 --- The Maximum-Likelihood Sequence Detector --- p.38Chapter 3.3.2 --- Maximum A Posteriori Probability Detector --- p.41Chapter 4. --- Multicode CDMA System Combined with Error-Correcting Codes --- p.45Chapter 4.1 --- Hamming Codes --- p.46Chapter 4.2 --- Gallager's Codes --- p.48Chapter 4.2.1 --- Encoding of Gallager's Codes --- p.48Chapter 4.2.2 --- Multicode Scheme combined with Gallager's Code --- p.52Chapter 4.2.3 --- Iterative Decoding of the Multicode Scheme --- p.55Chapter 4.3 --- Zigzag Codes --- p.59Chapter 4.4 --- Simulation Results and Discussion --- p.62Chapter 5. --- Multicode CDMA System with Bounded PAP Transmission --- p.68Chapter 5.1 --- Quantized Multicode Scheme --- p.69Chapter 5.1.1 --- System Model --- p.69Chapter 5.1.2 --- Interference of Code Channels --- p.71Chapter 5.2 --- Parallel Multicode Scheme --- p.74Chapter 5.2.1 --- System Model --- p.74Chapter 5.2.2 --- Selection of Hadamard Code Sequences --- p.75Chapter 6. --- Conclusions and Future Work --- p.82Chapter 6.1 --- Conclusions --- p.82Chapter 6.2 --- Future Work --- p.84Bibliography --- p.8

    Efficient fault-tolerant quantum computing

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    Fault tolerant quantum computing methods which work with efficient quantum error correcting codes are discussed. Several new techniques are introduced to restrict accumulation of errors before or during the recovery. Classes of eligible quantum codes are obtained, and good candidates exhibited. This permits a new analysis of the permissible error rates and minimum overheads for robust quantum computing. It is found that, under the standard noise model of ubiquitous stochastic, uncorrelated errors, a quantum computer need be only an order of magnitude larger than the logical machine contained within it in order to be reliable. For example, a scale-up by a factor of 22, with gate error rate of order 10510^{-5}, is sufficient to permit large quantum algorithms such as factorization of thousand-digit numbers.Comment: 21 pages plus 5 figures. Replaced with figures in new format to avoid problem

    High-Rate Regenerating Codes Through Layering

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    In this paper, we provide explicit constructions for a class of exact-repair regenerating codes that possess a layered structure. These regenerating codes correspond to interior points on the storage-repair-bandwidth tradeoff, and compare very well in comparison to scheme that employs space-sharing between MSR and MBR codes. For the parameter set (n,k,d=k)(n,k,d=k) with n<2k1n < 2k-1, we construct a class of codes with an auxiliary parameter ww, referred to as canonical codes. With ww in the range nk<w<kn-k < w < k, these codes operate in the region between the MSR point and the MBR point, and perform significantly better than the space-sharing line. They only require a field size greater than w+nkw+n-k. For the case of (n,n1,n1)(n,n-1,n-1), canonical codes can also be shown to achieve an interior point on the line-segment joining the MSR point and the next point of slope-discontinuity on the storage-repair-bandwidth tradeoff. Thus we establish the existence of exact-repair codes on a point other than the MSR and the MBR point on the storage-repair-bandwidth tradeoff. We also construct layered regenerating codes for general parameter set (n,k<d,k)(n,k<d,k), which we refer to as non-canonical codes. These codes also perform significantly better than the space-sharing line, though they require a significantly higher field size. All the codes constructed in this paper are high-rate, can repair multiple node-failures and do not require any computation at the helper nodes. We also construct optimal codes with locality in which the local codes are layered regenerating codes.Comment: 20 pages, 9 figure
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