Symbol Message Passing Decoding of Nonbinary Low-Density Parity-Check Codes

We present a novel decoding algorithm for q-ary low-density parity-check codes, termed symbol message passing. The proposed algorithm can be seen as a generalization of Gallager B and the binary message passing algorithm by Lechner et al. to q-ary codes. We derive density evolution equations for the q-ary symmetric channel, compute thresholds for a number of regular low-density parity-check code ensembles, and verify those by Monte Carlo simulations of long channel codes. The proposed algorithm shows performance advantages with respect to an algorithm of comparable complexity from the literature

Low-density parity-check codes with rates very close to the capacity of the q-ary symmetric channel for large q

Transmission of packets over computer networks is subject to packet-level errors, which appear as "bursts" of bit-level errors and are not well modeled by memoryless binary channels. A standard scrambling technique is used for transmission of packets by the q-ary symmetric channel (q-SC) with alphabet size q and error probability p. Furthermore, significant improvements in computational efficiency can be obtained by codes that operate at the packet-level instead of the bit-level. The capacity of the q-SC is 1+p log/sub q/ (p)+(1-p)log/sub q/ (1-p)-plog/sub q/ (1-q), which is close to 1-p for large q. First designed an efficient decoding algorithm for LDPC codes on the q-SC, and showed that it can afford rates arbitrarily close to 1-2p. We improve the analysis of this decoding algorithm to show that LDPC codes with the Tornado edge distribution, together with an erasure precode, can achieve rates e-close to capacity. We also extend this decoder into a family of decoding algorithms, which become progressively more powerful but also more complex. We show that in the limit, when the decoder is allowed to look infinitely deep into the decoding tree, it can achieve capacity without precoding. However computational requirements make this decoder impractical

LDPC Codes over Large Alphabets and Their Applications to Compressed Sensing and Flash Memory

This dissertation is mainly focused on the analysis, design and optimization of Low-density parity-check (LDPC) codes over channels with large alphabet sets and the applications on compressed sensing (CS) and flash memories. Compared to belief-propagation (BP) decoding, verification-based (VB) decoding has significantly lower complexity and near optimal performance when the channel alphabet set is large. We analyze the verification-based decoding of LDPC codes over the q-ary symmetric channel (q-SC) and propose the list-message-passing (LMP) decoding which off ers a good tradeoff between complexity and decoding threshold. We prove that LDPC codes with LMP decoding achieve the capacity of the q-SC when q and the block length go to infinity. CS is a newly emerging area which is closely related to coding theory and information theory. CS deals with the sparse signal recovery problem with small number of linear measurements. One big challenge in CS literature is to reduce the number of measurements required to reconstruct the sparse signal. In this dissertation, we show that LDPC codes with verification-based decoding can be applied to CS system with surprisingly good performance and low complexity. We also discuss modulation codes and error correcting codes (ECC’s) design for flash memories. We design asymptotically optimal modulation codes and discuss their improvement by using the idea from load-balancing theory. We also design LDPC codes over integer rings and fields with large alphabet sets for flash memories