Quantized Iterative Message Passing Decoders with Low Error Floor for LDPC Codes

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 -- variously referred to as near codewords, trapping sets, absorbing sets, or pseudocodewords -- 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. Using a theoretical analysis of iterative MP decoding in an idealized trapping set scenario, we show that a contributor to the error floors observed in the literature may be the imprecise implementation of decoding algorithms and, in particular, the message quantization rules used. We then propose a new quantization method -- (q+1)-bit quasi-uniform quantization -- that efficiently increases the dynamic range of messages, thereby overcoming a limitation of conventional quantization schemes. Finally, we use the quasi-uniform quantizer to decode several LDPC codes that suffer from high error floors with traditional fixed-point decoder implementations. The performance simulation results provide evidence that the proposed quantization scheme can, for a wide variety of codes, significantly lower error floors with minimal increase in decoder complexity

The approximate maximum-likelihood certificate

A new property which relies on the linear programming (LP) decoder, the approximate maximum-likelihood certificate (AMLC), is introduced. When using the belief propagation decoder, this property is a measure of how close the decoded codeword is to the LP solution. Using upper bounding techniques, it is demonstrated that the conditional frame error probability given that the AMLC holds is, with some degree of confidence, below a threshold. In channels with low noise, this threshold is several orders of magnitude lower than the simulated frame error rate, and our bound holds with very high degree of confidence. In contrast, showing this error performance by simulation would require very long Monte Carlo runs. When the AMLC holds, our approach thus provides the decoder with extra error detection capability, which is especially important in applications requiring high data integrity

Distributed Arithmetic Coding for the Asymmetric Slepian-Wolf problem

Distributed source coding schemes are typically based on the use of channels codes as source codes. In this paper we propose a new paradigm, termed "distributed arithmetic coding", which exploits the fact that arithmetic codes are good source as well as channel codes. In particular, we propose a distributed binary arithmetic coder for Slepian-Wolf coding with decoder side information, along with a soft joint decoder. The proposed scheme provides several advantages over existing Slepian-Wolf coders, especially its good performance at small block lengths, and the ability to incorporate arbitrary source models in the encoding process, e.g. context-based statistical models. We have compared the performance of distributed arithmetic coding with turbo codes and low-density parity-check codes, and found that the proposed approach has very competitive performance.Comment: submitted to IEEE Transactions on Signal processing, Nov. 2007. Revised version accepted with minor revision

Serial Concatenation of RS Codes with Kite Codes: Performance Analysis, Iterative Decoding and Design

In this paper, we propose a new ensemble of rateless forward error correction (FEC) codes. The proposed codes are serially concatenated codes with Reed-Solomon (RS) codes as outer codes and Kite codes as inner codes. The inner Kite codes are a special class of prefix rateless low-density parity-check (PRLDPC) codes, which can generate potentially infinite (or as many as required) random-like parity-check bits. The employment of RS codes as outer codes not only lowers down error-floors but also ensures (with high probability) the correctness of successfully decoded codewords. In addition to the conventional two-stage decoding, iterative decoding between the inner code and the outer code are also implemented to improve the performance further. The performance of the Kite codes under maximum likelihood (ML) decoding is analyzed by applying a refined Divsalar bound to the ensemble weight enumerating functions (WEF). We propose a simulation-based optimization method as well as density evolution (DE) using Gaussian approximations (GA) to design the Kite codes. Numerical results along with semi-analytic bounds show that the proposed codes can approach Shannon limits with extremely low error-floors. It is also shown by simulation that the proposed codes performs well within a wide range of signal-to-noise-ratios (SNRs).Comment: 34 pages, 15 figure

Controlling the Error Floor in LDPC Decoding

The error floor of LDPC is revisited as an effect of dynamic message behavior in the so-called absorption sets of the code. It is shown that if the signal growth in the absorption sets is properly balanced by the growth of set-external messages, the error floor can be lowered to essentially arbitrarily low levels. Importance sampling techniques are discussed and used to verify the analysis, as well as to discuss the impact of iterations and message quantization on the code performance in the ultra-low BER (error floor) regime.Comment: 11 pages, 7 figures, Submitted to IEEE Trans. Com

Block Markov Superposition Transmission of BCH Codes with Iterative Erasures-and-Errors Decoders

In this paper, we present the block Markov superposition transmission of BCH (BMST-BCH) codes, which can be constructed to obtain a very low error floor. To reduce the implementation complexity, we design a low complexity iterative sliding-window decoding algorithm, in which only binary and/or erasure messages are processed and exchanged between processing units. The error floor can be predicted by a genie-aided lower bound, while the waterfall performance can be analyzed by the density evolution method. To evaluate the error floor of the constructed BMST-BCH codes at a very low bit error rate (BER) region, we propose a fast simulation approach. Numerical results show that, at a target BER of $10^{-15}$, the hard-decision decoding of the BMST-BCH codes with overhead $25\%$ can achieve a net coding gain (NCG) of $10.55$ dB. Furthermore, the soft-decision decoding can yield an NCG of $10.74$ dB. The construction of BMST-BCH codes is flexible to trade off latency against performance at all overheads of interest and may find applications in optical transport networks as an attractive~candidate.Comment: submitted to IEEE Transactions on Communication

Rolex: Resilience-Oriented Language Extensions for Extreme-Scale Systems

Future exascale high-performance computing (HPC) systems will be constructed from VLSI devices that will be less reliable than those used today, and faults will become the norm, not the exception. This will pose significant problems for system designers and programmers, who for half-a-century have enjoyed an execution model that assumed correct behavior by the underlying computing system. The mean time to failure (MTTF) of the system scales inversely to the number of components in the system and therefore faults and resultant system level failures will increase, as systems scale in terms of the number of processor cores and memory modules used. However every error detected need not cause catastrophic failure. Many HPC applications are inherently fault resilient. Yet it is the application programmers who have this knowledge but lack mechanisms to convey it to the system. In this paper, we present new Resilience Oriented Language Extensions (Rolex) which facilitate the incorporation of fault resilience as an intrinsic property of the application code. We describe the syntax and semantics of the language extensions as well as the implementation of the supporting compiler infrastructure and runtime system. Our experiments show that an approach that leverages the programmer's insight to reason about the context and significance of faults to the application outcome significantly improves the probability that an application runs to a successful conclusion

Linear code-based vector quantization for independent random variables

In this paper we analyze the rate-distortion function R(D) achievable using linear codes over GF(q), where q is a prime number.Comment: 16 pages, 3 figure

Exhausting Error-Prone Patterns in LDPC Codes

It is proved in this work that exhaustively determining bad patterns in arbitrary, finite low-density parity-check (LDPC) codes, including stopping sets for binary erasure channels (BECs) and trapping sets (also known as near-codewords) for general memoryless symmetric channels, is an NP-complete problem, and efficient algorithms are provided for codes of practical short lengths n~=500. By exploiting the sparse connectivity of LDPC codes, the stopping sets of size <=13 and the trapping sets of size <=11 can be efficiently exhaustively determined for the first time, and the resulting exhaustive list is of great importance for code analysis and finite code optimization. The featured tree-based narrowing search distinguishes this algorithm from existing ones for which inexhaustive methods are employed. One important byproduct is a pair of upper bounds on the bit-error rate (BER) & frame-error rate (FER) iterative decoding performance of arbitrary codes over BECs that can be evaluated for any value of the erasure probability, including both the waterfall and the error floor regions. The tightness of these upper bounds and the exhaustion capability of the proposed algorithm are proved when combining an optimal leaf-finding module with the tree-based search. These upper bounds also provide a worst-case-performance guarantee which is crucial to optimizing LDPC codes for extremely low error rate applications, e.g., optical/satellite communications. Extensive numerical experiments are conducted that include both randomly and algebraically constructed LDPC codes, the results of which demonstrate the superior efficiency of the exhaustion algorithm and its significant value for finite length code optimization.Comment: submitted to IEEE Trans. Information Theor

Joint Decoding of LDPC Codes and Finite-State Channels via Linear-Programming

This paper considers the joint-decoding (JD) problem for finite-state channels (FSCs) and low-density parity-check (LDPC) codes. In the first part, the linear-programming (LP) decoder for binary linear codes is extended to JD of binary-input FSCs. In particular, we provide a rigorous definition of LP joint-decoding pseudo-codewords (JD-PCWs) that enables evaluation of the pairwise error probability between codewords and JD-PCWs in AWGN. This leads naturally to a provable upper bound on decoder failure probability. If the channel is a finite-state intersymbol interference channel, then the joint LP decoder also has the maximum-likelihood (ML) certificate property and all integer-valued solutions are codewords. In this case, the performance loss relative to ML decoding can be explained completely by fractional-valued JD-PCWs. After deriving these results, we discovered some elements were equivalent to earlier work by Flanagan on LP receivers. In the second part, we develop an efficient iterative solver for the joint LP decoder discussed in the first part. In particular, we extend the approach of iterative approximate LP decoding, proposed by Vontobel and Koetter and analyzed by Burshtein, to this problem. By taking advantage of the dual-domain structure of the JD-LP, we obtain a convergent iterative algorithm for joint LP decoding whose structure is similar to BCJR-based turbo equalization (TE). The result is a joint iterative decoder whose per-iteration complexity is similar to that of TE but whose performance is similar to that of joint LP decoding. The main advantage of this decoder is that it appears to provide the predictability of joint LP decoding and superior performance with the computational complexity of TE. One expected application is coding for magnetic storage where the required block-error rate is extremely low and system performance is difficult to verify by simulation.Comment: Accepted to IEEE Journal of Selected Topics in Signal Processing (Special Issue on Soft Detection for Wireless Transmission