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36
Design of LDPC Decoders for Improved Low Error Rate Performance: Quantization and Algorithm Choices
, 2009
"... Many classes of high-performance low-density parity-check (LDPC) codes are based on parity check matrices composed of permutation submatrices. We describe the design of a parallel-serial decoder architecture that can be used to map any LDPC code with such a structure to a hardware emulation platfor ..."
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Cited by 26 (10 self)
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Many classes of high-performance low-density parity-check (LDPC) codes are based on parity check matrices composed of permutation submatrices. We describe the design of a parallel-serial decoder architecture that can be used to map any LDPC code with such a structure to a hardware emulation platform. High-throughput emulation allows for the exploration of the low bit-error rate (BER) region and provides statistics of the error traces, which illuminate the causes of the error floors of the (2048, 1723) Reed-Solomon based LDPC (RS-LDPC) code and the (2209, 1978) array-based LDPC code. Two classes of error events are observed: oscillatory behavior and convergence to a class of non-codewords, termed absorbing sets. The influence of absorbing sets can be exacerbated by message quantization and decoder implementation. In particular, quantization and the log-tanh function approximation in sum-product decoders
Lowering LDPC Error Floors by Postprocessing
"... Abstract−A class of combinatorial structures, called absorbing sets, strongly influences the performance of low-density paritycheck (LDPC) decoders at low error rates. Past experiments have shown that a class of (8,8) absorbing sets determines the error floor performance of the (2048,1723) Reed-Solo ..."
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Cited by 22 (7 self)
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Abstract−A class of combinatorial structures, called absorbing sets, strongly influences the performance of low-density paritycheck (LDPC) decoders at low error rates. Past experiments have shown that a class of (8,8) absorbing sets determines the error floor performance of the (2048,1723) Reed-Solomon based LDPC code (RS-LDPC). A postprocessing approach is formulated to exploit the structure of the absorbing set by biasing the reliabilities of selected messages in a message-passing decoder. The approach converges quickly and can be efficiently implemented with minimal overhead. Hardware emulation of the decoder with postprocessing shows more than two orders of magnitude improvement in the very low bit error rate performance and errorfloor-free operation below a BER of 10-12. I.
Controlling LDPC Absorbing Sets via the Null Space of the Cycle Consistency Matrix
- In Proc. IEEE Int. Conf. on Comm. (ICC
, 2011
"... Abstract — This paper focuses on controlling absorbing sets for a class of regular LDPC codes, known as separable, circulantbased (SCB) codes. For a specified circulant matrix, SCB codes all share a common mother matrix and include array-based LDPC codes and many common quasi-cyclic codes. SCB codes ..."
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Cited by 11 (8 self)
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Abstract — This paper focuses on controlling absorbing sets for a class of regular LDPC codes, known as separable, circulantbased (SCB) codes. For a specified circulant matrix, SCB codes all share a common mother matrix and include array-based LDPC codes and many common quasi-cyclic codes. SCB codes retain standard properties of quasi-cyclic LDPC codes such as girth, code structure, and compatibility with existing high-throughput hardware implementations. This paper uses a cycle consistency matrix (CCM) for each absorbing set of interest in an SCB LDPC code. For an absorbing set to be present in an SCB LDPC code, the associated CCM must not be full columnrank. Our approach selects rows and columns from the SCB mother matrix to systematically eliminate dominant absorbing sets by forcing the associated CCMs to be full column-rank. Simulation results demonstrate that the new codes have steeper error-floor slopes and provide at least one order of magnitude of improvement in the low FER region. Identifying absorbingset-spectrum equivalence classes within the family of SCB codes with a specified circulant matrix significantly reduces the search space of possible code matrices. I.
Predicting Error Floors of Structured LDPC Codes: Deterministic Bounds and Estimates
, 2009
"... The error-correcting performance of low-density parity check (LDPC) codes, when decoded using practical iterative decoding algorithms, is known to be close to Shannon limits for codes with suitably large blocklengths. A substantial limitation to the use of finite-length LDPC codes is the presence o ..."
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Cited by 9 (1 self)
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The error-correcting performance of low-density parity check (LDPC) codes, when decoded using practical iterative decoding algorithms, is known to be close to Shannon limits for codes with suitably large blocklengths. A substantial limitation to the use of finite-length LDPC codes is the presence of an error floor in the low frame error rate (FER) region. This paper develops a deterministic method of predicting error floors, based on high signal-to-noise ratio (SNR) asymptotics, applied to absorbing sets within structured LDPC codes. The approach is illustrated using a class of array-based LDPC codes, taken as exemplars of high-performance structured LDPC codes. The results are in very good agreement with a stochastic method based on importance sampling which, in turn, matches the hardwarebased experimental results. The importance sampling scheme uses a mean-shifted version of the original Gaussian density, appropriately centered between a codeword and a dominant absorbing set, to produce an unbiased estimator of the FER with substantial computational savings over a standard Monte Carlo estimator. Our deterministic estimates are guaranteed to be a lower bound to the error probability in the high SNR regime and extends the prediction of the error probability to as low as 10 −30. By adopting a channel-independent viewpoint, the usefulness of these results is demonstrated for both the standard Gaussian channel and a channel with mixture noise.
Quantized min-sum decoders with low error floor for LDPC codes
- in Proc. IEEE Int. Symp. on Inform. Theory
, 2012
"... Abstract—The error floor phenomenon observed with LDPC codes and their graph-based, iterative, message-passing (MP) decoders is commonly attributed to the existence of error-prone substructures in a Tanner graph representation of the code. Many approaches have been proposed to lower the error floor ..."
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Cited by 8 (5 self)
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Abstract—The error floor phenomenon observed with LDPC codes and their graph-based, iterative, message-passing (MP) decoders is commonly attributed to the existence of error-prone substructures in a Tanner graph representation of the code. Many approaches have been proposed to lower the error floor by designing new LDPC codes with fewer such substructures or by modifying the decoding algorithm. In this paper, we show that one source of the error floors observed in the literature may be the message quantization rule used in the iterative decoder implementation. We then propose a new quantization method to overcome the limitations of standard quantization rules. Performance simulation results for two LDPC codes commonly found to have high error floors when used with the fixed-point min-sum decoder and its variants demonstrate the validity of our findings and the effectiveness of the proposed quantization algorithm. I.
LDPC Absorbing Sets, the Null Space of the Cycle Consistency Matrix, and Tanners Constructions
- In IEEE Conf. on Info. Theory and its Appl
, 2011
"... Abstract—Dolecek et al. introduced the cycle consistency condition, which is a necessary condition for cycles – and thus the absorbing sets that contain them – to be present in separable circulant-based (SCB) LDPC codes. This paper introduces a cycle consistency matrix (CCM) for each possible absorb ..."
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Cited by 7 (7 self)
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Abstract—Dolecek et al. introduced the cycle consistency condition, which is a necessary condition for cycles – and thus the absorbing sets that contain them – to be present in separable circulant-based (SCB) LDPC codes. This paper introduces a cycle consistency matrix (CCM) for each possible absorbing set in an SCB LDPC code. The CCM efficiently enforces the cycle consistency condition for all cycles in a specified absorbing set by spanning its associated binary cycle space. Under certain conditions, a CCM not having full column rank is a necessary and sufficient condition for the LDPC code to contain the absorbing set associated with that CCM. This paper uses the CCM approach to carefully analyze LDPC codes based on the Tanner construction for r = 4 rows of sub-matrices (i.e., Tannerconstruction LDPC codes with column weight 4). I.
Towards Improved LDPC Code Designs Using Absorbing Set Spectrum Properties
- In Proc. of 6th International symposium on
, 2010
"... Abstract—This paper focuses on methods for a systematic modification of the parity check matrix of regular LDPC codes for improved performance in the low BER region (i.e., the error floor). A judicious elimination of dominant absorbing sets strictly improves the absorbing set spectrum and thereby im ..."
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Cited by 6 (5 self)
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Abstract—This paper focuses on methods for a systematic modification of the parity check matrix of regular LDPC codes for improved performance in the low BER region (i.e., the error floor). A judicious elimination of dominant absorbing sets strictly improves the absorbing set spectrum and thereby improves the code performance. This absorbing set elimination is accomplished without compromising code properties and parameters such as the girth, node degree, and the structure of the parity check matrix. For a representative class of practical codes we substantiate theoretical analysis with experimental results obtained in the low BER region. Our results demonstrate at least an order of magnitude improvement of the error floor relative to the original code designs. Given that the conventional code parameters remain intact, the new code can easily be implemented on the existing software or hardware platforms employing high-throughput, compact architectures. As such, the proposed approach provides a step towards the improved code design that is compatible with practical implementation constraints. I.
Efficient Algorithms to Find All Small Error-Prone Substructures in LDPC Codes
- in Proc. IEEE Global Commun. Conf. (Globecom
"... Abstract—Exhaustively enumerating all small error-prone substructures in arbitrary, finite-length low-density parity-check (LDPC) codes has been proven to be NP-complete. In this paper, we present two exhaustive search algorithms to find such small error-prone substructures of an arbitrary LDPC code ..."
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Cited by 5 (4 self)
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Abstract—Exhaustively enumerating all small error-prone substructures in arbitrary, finite-length low-density parity-check (LDPC) codes has been proven to be NP-complete. In this paper, we present two exhaustive search algorithms to find such small error-prone substructures of an arbitrary LDPC code given its parity-check matrix. One algorithm is guaranteed to find all error-prone substructures including stopping sets, trapping sets, and absorbing sets, which have no more than amax variable nodes and up to bmax induced odd-degree neighboring check nodes. The other algorithm is specially designed to find fully absorbing sets (FAS). Numerical results show that both of our proposed algorithms are more efficient in terms of execution time than another recently proposed exhaustive search algorithm [13]. Moreover, by properly initialization of the algorithm, the efficiency can be further improved for quasi-cyclic (QC) codes. Index Terms—Low-density parity-check (LDPC) codes, exhaustive search, branch-and-bound, error floors, trapping sets. I.
On absorbing sets of structured sparse graph codes,” presented at the Inf
- Theory Appl. Workshop
, 2010
"... Abstract—In contrast to the capacity approaching performance of iteratively decoded low-density parity check (LDPC) codes, many practical finite-length LDPC codes exhibit performance degradation, manifested in a so-called error floor. Previous work has linked this phenomenon to the presence of certa ..."
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Cited by 4 (3 self)
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Abstract—In contrast to the capacity approaching performance of iteratively decoded low-density parity check (LDPC) codes, many practical finite-length LDPC codes exhibit performance degradation, manifested in a so-called error floor. Previous work has linked this phenomenon to the presence of certain combinatorial structures within the Tanner graph representation of the code, termed absorbing sets. Absorbing sets are stable under the bit-flipping operations and have been shown to act as fixed points (“absorbers”) for a wider class of iterative decoding algorithms. Codes often possess absorbing sets whose size is smaller than the minimum distance: the smallest absorbing sets are deemed most detrimental culprits behind the error floor. This paper focuses on the elementary combinatorial bounds of the smallest (candidate) absorbing sets. For certain classes of practical codes we demonstrate the tightness of these bounds and show how can the structure of the code and the structure of the absorbing sets be utilized to increase the size of the smallest absorbing sets without compromising other code properties such as the node degrees and the girth. As such, this work provides a step towards a better code design by taking into account the combinatorial nature of fixed points of iterative decoding algorithms. I.
Cyclic and quasi-cyclic LDPC codes on row and column constrained parity-check matrices and their trapping sets
- IEEE Trans. Inform. Theory
, 2012
"... ar ..."
