Multi-stage decoding of LDPC codes

Abstract

In this paper we present a three-stage decoding strategy that combines quantized and unquantized belief propagation (BP) decoders with a mixed-integer linear programming (MILP) decoder. Each decoding stage is activated only when the preceeding stage fails to converge to a valid codeword. The faster BP decoding stages are able to correct most errors, yielding a short average decoding time. Only in the rare cases when the iterative stages fail is the slower but more powerful MILP decoder used. The MILP decoder iteratively adds binary constraints until either the maximum likelihood codeword is found or some maximum number of binary constraints has been added. Simulation results demonstrate a large improvement in the word error rate (WER) of the proposed multi-stage decoder in comparison to belief propagation. The improvement is particularly noticeable in the low crossover probability (error floor) regime. Through introduction of an accelerated "active-set" version of the quantized BP decoder we significantly speed up the pace of simulation to simulate low density parity check (LDPC) codes of length up to around 2000 down to a WER of around 10(10) on the binary symmetric channel. We demonstrate that for certain codes our approach can efficiently approach the optimal ML decoding performance for low crossover probabilities. IEEE International Symposium on Information TheoryThis work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Abstract-In this paper we present a three-stage decoding strategy that combines quantized and un-quantized belief propagation (BP) decoders with a mixed-integer linear programming (MILP) decoder. Each decoding stage is activated only when the preceding stage fails to converge to a valid codeword. The faster BP decoding stages are able to correct most errors, yielding a short average decoding time. Only in the rare cases when the iterative stages fail is the slower but more powerful MILP decoder used. The MILP decoder iteratively adds binary constraints until either the maximum likelihood codeword is found or some maximum number of binary constraints has been added.Simulation results demonstrate a large improvement in the word error rate (WER) of the proposed multi-stage decoder in comparison to belief propagation. The improvement is particularly noticeable in the low crossover probability (error floor) regime. Through introduction of an accelerated "active-set" version of ...

Construction of high-girth QC-LDPC codes

“…1 Our initial experiments indeed show that girth-10 codes have a lower error floor compared with girth-6 and girth-8 codes of the same length and rate. This result, and other details about the codes found using our algorithm, will be discussed in depth in a future paper [19].…”

Decomposition Methods for Large Scale LP Decoding

“…Feldman and his collaborators showed that, for binary codes used over symmetric channels, a relaxed version of the maximum likelihood (ML) decoding problem can be stated as a linear program (LP). Considering graph-based 1 low-density parity-check (LDPC) codes, work by Feldman et al and later authors [4] [5] [6] [7] demonstrates that the bit-error-rate performance of LP decoding is competitive with that of standard sum-product (and min-sum) belief propagation (BP) decoding. Furthermore, LP decoding comes with a certificate of correctness (ML certificate) [3] -verifying with probability one when the decoder has found the ML codeword.…”

Finite Alphabet Iterative Decoders—Part I: Decoding Beyond Belief Propagation on the Binary Symmetric Channel

“…Some of the notable works include augmented belief propagation [17], informed dynamic scheduling [18], multi-stage decoding [19], averaged decoding [20], and the use of post-processing to lower the error floors [21], [22]. While all these schemes certainly provide performance enhancements in the error floor, all of them require either a considerable increase in decoding complexity due to the modifications and post-processing or are restricted to a particular code whose structure is well known.…”