3,727 research outputs found

    Deriving Good LDPC Convolutional Codes from LDPC Block Codes

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    Low-density parity-check (LDPC) convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity. In this paper we discuss several graph-cover-based methods for deriving families of time-invariant and time-varying LDPC convolutional codes from LDPC block codes and show how earlier proposed LDPC convolutional code constructions can be presented within this framework. Some of the constructed convolutional codes significantly outperform the underlying LDPC block codes. We investigate some possible reasons for this "convolutional gain," and we also discuss the --- mostly moderate --- decoder cost increase that is incurred by going from LDPC block to LDPC convolutional codes.Comment: Submitted to IEEE Transactions on Information Theory, April 2010; revised August 2010, revised November 2010 (essentially final version). (Besides many small changes, the first and second revised versions contain corrected entries in Tables I and II.

    Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms

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    Mathematical programming is a branch of applied mathematics and has recently been used to derive new decoding approaches, challenging established but often heuristic algorithms based on iterative message passing. Concepts from mathematical programming used in the context of decoding include linear, integer, and nonlinear programming, network flows, notions of duality as well as matroid and polyhedral theory. This survey article reviews and categorizes decoding methods based on mathematical programming approaches for binary linear codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory. Published July 201

    Error-Correcting Codes for Automatic Control

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    Systems with automatic feedback control may consist of several remote devices, connected only by unreliable communication channels. It is necessary in these conditions to have a method for accurate, real-time state estimation in the presence of channel noise. This problem is addressed, for the case of polynomial-growth-rate state spaces, through a new type of error-correcting code that is online and computationally efficient. This solution establishes a constructive analog, for some applications in estimation and control, of the Shannon coding theorem

    Minimum Pseudoweight Analysis of 3-Dimensional Turbo Codes

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    In this work, we consider pseudocodewords of (relaxed) linear programming (LP) decoding of 3-dimensional turbo codes (3D-TCs). We present a relaxed LP decoder for 3D-TCs, adapting the relaxed LP decoder for conventional turbo codes proposed by Feldman in his thesis. We show that the 3D-TC polytope is proper and CC-symmetric, and make a connection to finite graph covers of the 3D-TC factor graph. This connection is used to show that the support set of any pseudocodeword is a stopping set of iterative decoding of 3D-TCs using maximum a posteriori constituent decoders on the binary erasure channel. Furthermore, we compute ensemble-average pseudoweight enumerators of 3D-TCs and perform a finite-length minimum pseudoweight analysis for small cover degrees. Also, an explicit description of the fundamental cone of the 3D-TC polytope is given. Finally, we present an extensive numerical study of small-to-medium block length 3D-TCs, which shows that 1) typically (i.e., in most cases) when the minimum distance dmind_{\rm min} and/or the stopping distance hminh_{\rm min} is high, the minimum pseudoweight (on the additive white Gaussian noise channel) is strictly smaller than both the dmind_{\rm min} and the hminh_{\rm min}, and 2) the minimum pseudoweight grows with the block length, at least for small-to-medium block lengths.Comment: To appear in IEEE Transactions on Communication

    Convolutional and tail-biting quantum error-correcting codes

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    Rate-(n-2)/n unrestricted and CSS-type quantum convolutional codes with up to 4096 states and minimum distances up to 10 are constructed as stabilizer codes from classical self-orthogonal rate-1/n F_4-linear and binary linear convolutional codes, respectively. These codes generally have higher rate and less decoding complexity than comparable quantum block codes or previous quantum convolutional codes. Rate-(n-2)/n block stabilizer codes with the same rate and error-correction capability and essentially the same decoding algorithms are derived from these convolutional codes via tail-biting.Comment: 30 pages. Submitted to IEEE Transactions on Information Theory. Minor revisions after first round of review

    Feedback Communication Systems with Limitations on Incremental Redundancy

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    This paper explores feedback systems using incremental redundancy (IR) with noiseless transmitter confirmation (NTC). For IR-NTC systems based on {\em finite-length} codes (with blocklength NN) and decoding attempts only at {\em certain specified decoding times}, this paper presents the asymptotic expansion achieved by random coding, provides rate-compatible sphere-packing (RCSP) performance approximations, and presents simulation results of tail-biting convolutional codes. The information-theoretic analysis shows that values of NN relatively close to the expected latency yield the same random-coding achievability expansion as with N=N = \infty. However, the penalty introduced in the expansion by limiting decoding times is linear in the interval between decoding times. For binary symmetric channels, the RCSP approximation provides an efficiently-computed approximation of performance that shows excellent agreement with a family of rate-compatible, tail-biting convolutional codes in the short-latency regime. For the additive white Gaussian noise channel, bounded-distance decoding simplifies the computation of the marginal RCSP approximation and produces similar results as analysis based on maximum-likelihood decoding for latencies greater than 200. The efficiency of the marginal RCSP approximation facilitates optimization of the lengths of incremental transmissions when the number of incremental transmissions is constrained to be small or the length of the incremental transmissions is constrained to be uniform after the first transmission. Finally, an RCSP-based decoding error trajectory is introduced that provides target error rates for the design of rate-compatible code families for use in feedback communication systems.Comment: 23 pages, 15 figure

    Design and Analysis of Time-Invariant SC-LDPC Convolutional Codes With Small Constraint Length

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    In this paper, we deal with time-invariant spatially coupled low-density parity-check convolutional codes (SC-LDPC-CCs). Classic design approaches usually start from quasi-cyclic low-density parity-check (QC-LDPC) block codes and exploit suitable unwrapping procedures to obtain SC-LDPC-CCs. We show that the direct design of the SC-LDPC-CCs syndrome former matrix or, equivalently, the symbolic parity-check matrix, leads to codes with smaller syndrome former constraint lengths with respect to the best solutions available in the literature. We provide theoretical lower bounds on the syndrome former constraint length for the most relevant families of SC-LDPC-CCs, under constraints on the minimum length of cycles in their Tanner graphs. We also propose new code design techniques that approach or achieve such theoretical limits.Comment: 30 pages, 5 figures, accepted for publication in IEEE Transactions on Communication
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