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arXiv:0705.1345 · IEEE Information Theory Workshop (ITW)

Degree Optimization and Stability Condition for the Min-Sum Decoder

Kapil Bhattad, Vishwambhar Rathi, Rudiger L. Urbanke

Published Sep 1, 2007• 6 citations

Abstract

The min-sum (MS) algorithm is arguably the second most fundamental algorithm in the realm of message passing due to its optimality (for a tree code) with respect to theblockerrorprobability [1]. There also seems to be a fundamental relationship of MS decoding with the linear programming decoder [2]. Despite its importance, its fundamental properties have not nearly been studied as well as those of the sum-product (also known as BP) algorithm. We address two questions related to the MS rule. First, we characterize the stability condition under MS decoding. It turns out to be essentially the same condition as under BP decoding. Second, we perform a degree distribution optimization. Contrary to the case of BP decoding, under MS decoding the thresholds of the best degree distributions for standard irregular LDPC ensembles are significantly bounded away from the Shannon threshold. More precisely, on the AWGN channel, for the best codes that we find, the gap to capacity is 1 dB for a rate 0.3 code and it is 0.4 dB when the rate is 0.9 (the gap decreases monotonically as we increase the rate). We also used the optimization procedure to design codes for modified MS algorithm where the output of the check node is scaled by a constant 1/alpha. For alpha = 1.25, we observed that the gap to capacity was lesser for the modified MS algorithm when compared with the MS algorithm. However, it was still quite large, varying from 0.75 dB to 0.2 dB for rates between 0.3 and 0.9. We conclude by posing what we consider to be the most important open questions related to the MS algorithm.

DOI: 10.1109/itw.2007.4313072

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In this paper, a modified normalized Min-Sum algorithm (MNMSA) is proposed for decoding irregular Low Density Parity Check Codes (LDPC) with non-uniform degree distribution. The proposed algorithm provides an efficient decoding scheme to enhance the error performance of an irregular LDPC code without increasing the hardware complexity. An efficient 6- bit quantization scheme is utilized in the proposed algorithm with optimally combined normalization and down scaling factors to resolve the magnitude overestimation issue which occurs during iterative message passing process between the nodes. The simulation results demonstrate that the proposed algorithm achieves good decoding performance in terms of Frame Error Rate (FER) and Bit Error Rate (BER) without the requirement of complex arithmetic calculations. Compared to other min- sum based decoding algorithms the decoding process of the proposed MNMSA requires only fewer decoding iterations over wide range of signal- to- noise ratio (SNR) to achieve comparable decoding performance to that of Belief Propagation (BP) algorithm without increasing the hardware complexity.

A check node degree-based modified min-sum algorithm which classifies the codes was developed to reduce the complexity and improve the performance of decoding algorithm for low-density parity-check(LDPC) codes.The algorithm corrects both the minimum and sub-minimum absolute values of incoming messages in the check nodes.Two correction factors are obtained by calculating the offset between the belief propagation(BP) algorithm and the min-sum algorithm based on the check node degree.Simulations show that the algorithm achieves better performance than the BP algorithm at high signal noise ratio.The algorithm is also less complex than the BP algorithm and provides a good tradeoff between performance and complexity for any check node degree distribution in the LDPC codes.

Low Density Parity Check Codes (LDPC) give groundbreaking performance which is known to approach Shannon's limits for sufficiently large block length. Historically and recently, LDPC have been known to give superior performance than concatenated coding. In the following paper, a proposal to modify the standard Min-Sum (MS) algorithm for decoding LDPC codes is presented. This is done by introduction of a difference to sum ratio factor, κ in the check to bit node updating process. The algorithm is further extended by implementing hard decision of the Bit-Flipping (BF) algorithm over the soft decision of MS algorithm. Simulation results demonstrate that the proposed algorithms are effective in imparting a better performance in terms of a lower bit error rate (BER) at medium to high signal to noise ratio (SNR) when compared to the traditional MS algorithm while adding fair amount of complexity.

Simulations have shown that the outputs of min-sum (MS) decoding generally behave in one of two ways: either the output vector eventually stabilizes at a codeword or it eventually cycles through a finite set of vectors that may include both codewords and non-codewords. The latter behavior has significantly contributed to the difficulty in studying the performance of this decoder. To overcome this problem, a new decoder, average min-sum (AMS), is proposed; this decoder outputs the average of the MS output vectors over a finite set of iterations. Simulations comparing MS, AMS, linear programming (LP) decoding, and maximum likelihood (ML) decoding are presented, illustrating the relative performances of each of these decoders. In general, MS and AMS have comparable word error rates; however, in the simulation of a code with large block length, AMS has a significantly lower bit error rate. Finally, AMS pseudocodewords are introduced and their relationship to graph cover and LP pseudocodewords is explored, with particular focus on the AMS pseudocodewords of regular LDPC codes and cycle codes.

The performance of low-density parity-check (LDPC) codes transmitted over a memoryless binary-input continuous output additive white Gaussian noise (AWGN) channel and decoded with quantized min-sum decoding is strongly influenced by the decoder's quantization scheme. This paper presents an efficient algorithm that determines the best uniform scalar quantizer for a particular code. To maximize performance, it is necessary to determine degree distributions that best match the characteristics of the quantized min-sum decoder. Toward this end, an iterative optimization framework that jointly optimizes the degree distributions and the quantizer is presented