Iterative Decoding Algorithm for LDPC Codes based on Difference Map Algorithm

Ludovic Danjean, David Declercq, Shiva Kumar Planjery +1 more

Recently new message passing decoders for LDPC codes, called finite alphabet iterative decoders (FAIDs) were proposed. The messages belong to a finite alphabet and the update functions are simple boolean maps different from the functions used for the belied propagation (BP) decoder. The maps can be chosen using the knowledge of potential trapping sets such that the decoders surpass the BP decoder in the error floor. In this paper, we address the issue of selecting good FAIDs which perform well in the error floor for column weight three codes. We introduce the notion of noisy trapping set which is a generalization based on analyzing the local dynamic behaviour of a given FAID on a trapping set. Using this notion as the core, we provide an iterative greedy algorithm that outputs a set of candidate FAIDs containing potentially good decoders for any given code. To illustrate the appliance of the methodology on several codes, we show that the set of candidate FAIDs contains particularly good FAIDs for different codes with different rates and lengths.

Tom Høholdt, JØrn Justesen

Jinghu Chen, M.P.C. Fossorier

Iterative decoding techniques have been receiving more and more attentions with the invention of turbo codes and the rediscovery of low-density parity-check (LDPC) codes. An important aspect in the study of iterative decoding is the tradeoff between decoding performance and complexities. For both LDPC codes and turbo codes, optimum decoding algorithms can provide very good performance. However, complicated operations are involved in the optimum decoding, and prohibit the wide applications of LDPC codes and turbo codes in the next generation digital communication and storage systems. Although there exist sub-optimum decoding algorithms for both LDPC codes and turbo codes, the decoding performance is degraded with the sub-optimum algorithms, and under some circumstances, the gap is very large. This research investigates the reduced complexity decoding algorithms of LDPC codes and turbo codes. For decoding LDPC codes, new algorithms, namely the normalized BP-based algorithm and the offset BP-based algorithm, are proposed. For these two reduced complexity algorithms, density evolution algorithms are derived, and are used to determine the best decoder parameters associated with each of the algorithms. Numerical results show that the new algorithms can achieve near optimum decoding performances for infinite code lengths, and simulation results reveal the same conclusion for short to medium code lengths. In addition to the advantage of low computational complexities, the two new algorithms are less subject to quantization errors and correlation effects than the optimum BP algorithm, and consequently are more suitable for hardware implementation. For a special kind of LDPC codes - the geometric LDPC codes, we propose the normalized APP-based algorithm, which is even more simplified yet still can achieve the near optimum performance. For decoding turbo codes, two new sub-optimum decoding algorithms are proposed. The first is the bi-directional soft-output Viterbi algorithm (bi-SOVA), which is based on utilizing a backward SOYA decoding in addition to the conventional forward one, and can achieve better performance than the uni-directional SOYA. The second is the normalized Max-Log-MAP algorithm, which improves the performance of the Max-Log-MAP decoding by scaling the soft outputs with some predetermined factors.

Minghua Liu, Lijun Zhang

David Burshtein, Gadi Miller

Hossein Pishro-Nik, Faramarz Fekri

This paper investigates decoding of low-density parity-check (LDPC) codes over the binary erasure channel (BEC). We study the iterative and maximum-likelihood (ML) decoding of LDPC codes on this channel. We derive bounds on the ML decoding of LDPC codes on the BEC. We then present an improved decoding algorithm. The proposed algorithm has almost the same complexity as the standard iterative decoding. However, it has better performance. Simulations show that we can decrease the error rate by several orders of magnitude using the proposed algorithm. We also provide some graph-theoretic properties of different decoding algorithms of LDPC codes over the BEC which we think are useful to better understand the LDPC decoding methods, in particular, for finite-length codes.