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International Conference on Information and Communication Technology Convergence (ICTC)

Density Evolution of the Bit-Flipping Decoding Algorithm of Regular Low-Density Parity-Check Codes

Myungin Kim, Jeongseok Ha

Published Oct 1, 2019• 1 citation
Publisher

Abstract

In this paper, we develop the density evolution of bit-flipping algorithm (BFA) for regular low-density parity-check (LDPC) codes and analyze the evolution of probability density for the bit-decision error. In contrast to the sum-product algorithm (SPA) and min-sum algorithm (MSA), the density evolution for BFA has remained untouched, which motivates this work. In the developed density evolution for BFA, we first introduce a state variable which is a function of bit-decision result and the flipping function value at each iteration. Then, the state probability is derived, with which the density of bit-decision error is obtained. To confirm the derivations, we design regular LDPC codes and compare the empirical evaluations of bit-error rates and the derived probability of bit-decision errors.

DOI
10.1109/ictc46691.2019.8939981

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In this paper, we develop the density evolution of bit-flipping algorithm (BFA) for regular low-density parity-check (LDPC) codes and analyze the evolution of probability density for the bit-decision error. In contrast to the sum-product algorithm (SPA) and min-sum algorithm (MSA), the density evolution for BFA has remained untouched, which motivates this work. In the developed density evolution for BFA, we first introduce a state variable which is a function of bit-decision result and the flipping function value at each iteration. Then, the state probability is derived, with which the density of bit-decision error is obtained. To confirm the derivations, we design regular LDPC codes and compare the empirical evaluations of bit-error rates and the derived probability of bit-decision errors.
In this work, we analytically derive the evolution of single-bit messages in the bit-flipping algorithm (BFA) for irregular low-density parity-check (LDPC) codes on the binary symmetric channel (BSC). In particular, the analysis tracks the probability distributions of the flipping functions and the bit-decision error at each iteration, which is possible by introducing a state variable at each variable node. The probability distributions are indirectly obtained by taking advantage of the state and state transition probabilities. The analysis will be confirmed by comparing analytic evaluations with simulation results. For the comparison, we implement LDPC codes at a finite length.
The Bit-Flipping (BF) algorithm is considered as a hard decoding method for LDPC codes. It is much simpler than the probabilistic methods like Sum Product Algorithm (SPA), and can be efficiently implemented by electronic circuits. In this paper, we propose a new Bit Flipping algorithm for Low-Density Parity-Check codes (LDPC) called Single Bit-Flipping (SBF). Compared to the Gallager Bit-Flipping algorithm, the proposed algorithm employs criteria of flipping single bit chosen carefully. This method eliminates the risk for flipping more than one bit at a time that can induce additional error bits and propagate the errors to the later iterations. We present some results obtained by applying this new method that provides a gain in performance in comparison to the standard Gallager Bit-Flipping with low complexity.
In this paper we present Low Density Parity Check decoding algorithm that assemble two different algorithms: Sum-Product and Bit-Flipping; we denote Bit Flipping-Sum Product Algorithm (BFSP). To reduce the bit error rate, we perform the Bit-Flipping algorithm after the Sum-Product algorithm. Simulation results over an additive white Gaussian channel show that the error performance of a LDPC codes with Bit Flipping-Sum Product decoding is within 0.2 dB of the standard Sum-Product decoding algorithms. Furthermore, the decoding complexity of the proposed BFSP algorithm is maintained at the same level as that of the standard Sum-Product algorithm.
It is proved that hard decision algorithms are more appropriate than a soft decision for low-density parity-check (LDPC) decoding since they are less complex at the decoding level. On the other hand, it is notable that the soft decision algorithm outperforms the hard decision one in terms of the bit error rate (BER) gap. In order to minimize the BER and the gap between these two families of LDPC codes, a new LDPC decoding algorithm is suggested in this paper, which is based on both the normalized min-sum (NMS) and modified-weighted bit-flipping (MWBF). The proposed algorithm is named normalized min sum- modified weighted bit flipping (NMSMWBF). The MWBF is executed after the NMS algorithm. The simulations show that our algorithm outperforms the NMS in terms of BER at 10-8 over the additive white gaussian noise (AWGN) channel by 0.25 dB. Furthermore, the proposed NMSMWBF and the NMS are both at the same level of decoding difficulty.
An improved bit-flipping decoding algorithm is proposed for low-density parity-check(LDPC) codes in this paper.Two influences on error probabilities of code bits are considered: the influence of the number of code bits included in an unsatisfied check,and the influence of the number of unsatisfied checks of each code bit.No soft information is required for decoding.Simulation results show that improvement on performance is observed for the proposed algorithm,and it provides better performance over several soft-decision bit-flipping decoding algorithms for LDPC codes with large girth.
In order to improve the bit-flipping (BF) decoding algorithms, a weighted hard-decision based BF decoding algorithm for low-density parity-check (LDPC) codes is presented, in which the weights of variable-nodes are employed. A new metric is defined for the weight of candidate bits not only in unsatisfied parity check equations but also in satisfied ones. The connection between min-sum decoding algorithm and the proposed algorithm is presented. Simulation results show that the new decoding algorithm can provide better FER performance and faster convergence speed than the original candidate bit based bit-flipping (CBBF) algorithm.