Iterative Algebraic Soft-Decision List Decoding of Reed-Solomon Codes

In this paper, we present an iterative soft-decision decoding algorithm for Reed-Solomon codes offering both complexity and performance advantages over previously known decoding algorithms. Our algorithm is a list decoding algorithm which combines two powerful soft decision decoding techniques which were previously regarded in the literature as competitive, namely, the Koetter-Vardy algebraic soft-decision decoding algorithm and belief-propagation based on adaptive parity check matrices, recently proposed by Jiang and Narayanan. Building on the Jiang-Narayanan algorithm, we present a belief-propagation based algorithm with a significant reduction in computational complexity. We introduce the concept of using a belief-propagation based decoder to enhance the soft-input information prior to decoding with an algebraic soft-decision decoder. Our algorithm can also be viewed as an interpolation multiplicity assignment scheme for algebraic soft-decision decoding of Reed-Solomon codes.Comment: Submitted to IEEE for publication in Jan 200

Density Evolution and Functional Threshold for the Noisy Min-Sum Decoder

This paper investigates the behavior of the Min-Sum decoder running on noisy devices. The aim is to evaluate the robustness of the decoder in the presence of computation noise, e.g. due to faulty logic in the processing units, which represents a new source of errors that may occur during the decoding process. To this end, we first introduce probabilistic models for the arithmetic and logic units of the the finite-precision Min-Sum decoder, and then carry out the density evolution analysis of the noisy Min-Sum decoder. We show that in some particular cases, the noise introduced by the device can help the Min-Sum decoder to escape from fixed points attractors, and may actually result in an increased correction capacity with respect to the noiseless decoder. We also reveal the existence of a specific threshold phenomenon, referred to as functional threshold. The behavior of the noisy decoder is demonstrated in the asymptotic limit of the code-length -- by using "noisy" density evolution equations -- and it is also verified in the finite-length case by Monte-Carlo simulation.Comment: 46 pages (draft version); extended version of the paper with same title, submitted to IEEE Transactions on Communication

New Identification and Decoding Techniques for Low-Density Parity-Check Codes

Error-correction coding schemes are indispensable for high-capacity high data-rate communication systems nowadays. Among various channel coding schemes, low-density parity-check (LDPC) codes introduced by pioneer Robert G. Gallager are prominent due to the capacity-approaching and superior error-correcting properties. There is no hard constraint on the code rate of LDPC codes. Consequently, it is ideal to incorporate LDPC codes with various code rate and codeword length in the adaptive modulation and coding (AMC) systems which change the encoder and the modulator adaptively to improve the system throughput. In conventional AMC systems, a dedicated control channel is assigned to coordinate the encoder/decoder changes. A questions then rises: if the AMC system still works when such a control channel is absent. This work gives positive answer to this question by investigating various scenarios consisting of different modulation schemes, such as quadrature-amplitude modulation (QAM), frequency-shift keying (FSK), and different channels, such as additive white Gaussian noise (AWGN) channels and fading channels. On the other hand, LDPC decoding is usually carried out by iterative belief-propagation (BP) algorithms. As LDPC codes become prevalent in advanced communication and storage systems, low-complexity LDPC decoding algorithms are favored in practical applications. In the conventional BP decoding algorithm, the stopping criterion is to check if all the parities are satisfied. This single rule may not be able to identify the undecodable blocks, as a result, the decoding time and power consumption are wasted for executing unnecessary iterations. In this work, we propose a new stopping criterion to identify the undecodable blocks in the early stage of the iterative decoding process. Furthermore, in the conventional BP decoding algorithm, the variable (check) nodes are updated in parallel. It is known that the number of iterations can be reduced by the serial scheduling algorithm. The informed dynamic scheduling (IDS) algorithms were proposed in the existing literatures to further reduce the number of iterations. However, the computational complexity involved in finding the update node in the existing IDS algorithms would not be neglected. In this work, we propose a new efficient IDS scheme which can provide better performance-complexity trade-off compared to the existing IDS ones. In addition, the iterative decoding threshold, which is used for differentiating which LDPC code is better, is investigated in this work. A family of LDPC codes, called LDPC convolutional codes, has drawn a lot of attentions from researchers in recent years due to the threshold saturation phenomenon. The IDT for an LDPC convolutional code may be computationally demanding when the termination length goes to thousand or even approaches infinity, especially for AWGN channels. In this work, we propose a fast IDT estimation algorithm which can greatly reduce the complexity of the IDT calculation for LDPC convolutional codes with arbitrary large termination length (including infinity). By utilizing our new IDT estimation algorithm, the IDTs for LDPC convolutional codes with arbitrary large termination length (including infinity) can be quickly obtained