Density Evolution for Nonbinary LDPC Codes Under Gaussian Approximation

• Published in: IEEE transactions on information theory Vol. 55; no. 3; pp. 997 - 1015
• Main Authors: Ge Li, Fair, I.J., Krzymien, W.A.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-03-2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Additive white noise; Applied sciences; Approximation; AWGN; Binary system; Channels; Codes; Coding, codes; Data encryption; Decoding; Density; density evolution; Design optimization; Evolution; Exact sciences and technology; Gaussian; Gaussian approximation; Information theory; Information, signal and communications theory; Linear programming; low-density parity-check (LDPC) codes; Mathematical analysis; Messages; Modulation coding; Modulation, demodulation; Noise; Normal distribution; Parity check codes; Phase modulation; Phase shift keying; Signal and communications theory; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Transmission and modulation (techniques and equipments); Performance evaluation; Binary channel; AWGN; density evolution; Linear programming; Decoding; Density distribution; Algorithm; Gaussian approximation; AWGN channels; Phase shift keying; Gaussian process; Binary modulation; low-density parity-check (LDPC) codes; Parity check codes; Error correcting code
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2008.2011435

This paper extends the work on density evolution for binary low-density parity-check (LDPC) codes with Gaussian approximation to LDPC codes over GF(q) . We first generalize the definition of channel symmetry for nonbinary inputs to include q-ary phase-shift keying (PSK) modulated channels for prime q and binary-modulated channels for q that is a power of 2. For the well-defined q-ary-input symmetric-output channel, we prove that under the Gaussian assumption, the density distribution for messages undergoing decoding is fully characterized by (q-1) quantities. Assuming uniform edge weights, we further show that the density of messages computed by the check node decoder (CND) is fully defined by a single number. We then present the approximate density evolution for regular and irregular LDPC codes, and show that the (q-1) -dimensional integration involved can be simplified using a dimensionality reduction algorithm for the important case of q=2 p . Through application of approximate density evolution and linear programming, we optimize the degree distribution of LDPC codes over GF(3) and GF(4). The optimized irregular LDPC codes demonstrate performance close to the Shannon capacity for long codewords. We also design GF(q) codes for high-order modulation by using the idea of a channel adapter. We find that codes designed in this fashion outperform those optimized specifically for the binary additive white Gaussian noise (AWGN) channel for a short codewords and a spectral efficiency of 2 bits per channel use (b/cu).