A modified Gradient Descent Bit Flipping (GDBF) algorithm is proposed for decoding Low Density Parity Check (LDPC) codes on the binary-input additive white Gaussian noise channel. The new algorithm, called Noisy GDBF (NGDBF), introduces a random perturbation into each symbol metric at each iteration. The noise perturbation allows the algorithm to escape from undesirable local maxima, resulting in improved performance. A combination of heuristic improvements to the algorithm are proposed and evaluated. When the proposed heuristics are applied, NGDBF performs better than any previously reported GDBF variant, and comes within 0.5 dB of the belief propagation algorithm for several tested codes. Unlike other previous GDBF algorithms that provide an escape from local maxima, the proposed algorithm uses only local, fully parallelizable operations and does not require computing a global objective function or a sort over symbol metrics, making it highly efficient in comparison. The proposed NGDBF algorithm requires channel state information which must be obtained from a signal to noise ratio (SNR) estimator. Architectural details are presented for implementing the NGDBF algorithm. Complexity analysis and optimizations are also discussed.
The original bit-flipping algorithm (BFA) was introduced by Gallager in his seminal work on LDPC codes @cite_8 . Gallager's BFA is a hard-decision algorithm for decoding on the binary symmetric channel (BSC), in which only hard channel bits are available to the decoder. To correct errors, the BFA computes a sum over the adjacent parity-check equations for each bit in the code. If, for any bit, the number of adjacent parity violations exceeds a specified threshold, then the bit is flipped. This process is repeated until all parity checks are satisfied, or until a maximum iteration limit is reached. The BFA has very low complexity since it only requires, in each iteration, a summation over binary parity-check values for each symbol; however the BFA provides weak decoding performance. considered a probabilistic BFA (PBFA) which adds randomness to the bit-flip decision, resulting in improved performance @cite_9 . In PBFA, when a bit's parity-check sum crosses the flip threshold, it is flipped with probability @math . The parameter @math is optimized empirically and is adapted toward 1 during successive iterations.
@cite_9 @cite_8
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In this correspondence, a new method for improving hard-decision bit-flipping decoding of low-density parity-check (LDPC) codes is presented. Bits with a number of unsatisfied check sums larger than a predetermined threshold are flipped with a probability p spl les 1 which is independent of the code considered. The probability p is incremented during decoding according to some rule. With a proper choice of the initial p, the proposed improved bit-flipping (BF) algorithm achieves gain not only in performance, but also in average decoding time for signal-to-noise ratio (SNR) values of interest with respect to p = 1. A novel class of bit-flipping (BF) algorithm for decoding low-density parity-check (LDPC) codes is presented. The proposed algorithms, which are referred to as gradient descent bit flipping (GDBF) algorithms, can be regarded as simplified gradient descent algorithms. The proposed algorithms exhibit better decoding performance than known BF algorithms, such as the weighted BF algorithm or the modified weighted BF algorithm for several LDPC codes. A parallel weighted bit-flipping (PWBF) decoding algorithm for low-density parity-check (LDPC) codes is proposed. Compared to the best known serial weighted bit-flipping decoding, the PWBF decoding converges significantly faster but with little performance penalty. For decoding of finite-geometry LDPC codes, we demonstrate through examples that the proposed PWBF decoding converges in about 5 iterations with performance very close to that of the standard belief-propagation decoding. We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous nonasymptotic results for LDPC codes, and in particular, exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a flow on the Tanner graph of the code. An interesting by-product of our analysis is to establish the existence of ldquoprobabilistic expansionrdquo in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst case setting.