IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009 997 Density Evolution for Nonbinary LDPC Codes Under Gaussian Approximation Ge Li, Ivan J. Fair, Member, IEEE, and Witold A. Krzymien´ , Senior Member, IEEE Abstract—This paper extends the work on density evolution for binary low-density parity-check (LDPC) codes with Gaussian approximation to LDPC codes over GF . We first generalize the definition of channel symmetry for nonbinary inputs to include -ary phase-shift keying (PSK) modulated channels for prime and binary-modulated channels for that is a power of . For the well-defined -ary-input symmetric-output channel, we prove that under the Gaussian assumption, the density distribution for messages undergoing decoding is fully characterized by 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 -dimensional integration involved can be simplified using a dimensionality reduction algorithm for the important case of . Through application of approximate density evolution and linear programming, we optimize the degree distribution of LDPC codes over GF and GF . The optimized irregular LDPC codes demonstrate performance close to the Shannon capacity for long codewords. We also design GF 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). Index Terms—density evolution, Gaussian approximation, low- density parity-check (LDPC) codes. I. INTRODUCTION T HIS paper is motivated by the impressive performance of irregular low-density parity-check (LDPC) codes over a wide class of channels [1]–[5]. Irregular LDPC codes are commonly designed using a numerical technique called density evolution. Developed by Richardson and Urbanke [1], the method of density evolution is one of the most powerful tools known for analyzing the asymptotic performance of an LDPC Manuscript received February 14, 2007; revised April 24, 2008. Current ver- sion published February 25, 2009. This work was supported by TRLabs, the Rohit Sharma Professorship, and the Natural Sciences and Engineering Re- search Council (NSERC) of Canada. The material in this paper was presented in part at the IEEE International Symposium on Information Theory, Yokohama, Japan, June 2003. G. Li was with the Department of Electrical and Computer Engineering/TR- Labs, University of Alberta, Edmonton AB T6G 2V4 , Canada. She is now with JP Morgan Securities Inc., New York, NY 10017 USA (e-mail: geli@ece.ual- berta.ca). I. J. Fair and W. A. Krzymien´ are with the Department of Electrical and Com- puter Engineering/TRLabs, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: fair@ece.ualberta.ca; wak@ece.ualberta.ca). Communicated by H.-A. Loeliger, Associate Editor for Coding Techniques. Color versions of Figures 2, 4–6, 8, and 9 in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIT.2008.2011435 code ensemble specified by a degree distribution pair. With the symmetry of the channel and decoding algorithm as one of its fundamental assumptions, this method can be applied to a wide class of binary-input symmetric-output channels, including the binary erasure channel (BEC), binary symmetric channel (BSC), and additive white Gaussian noise (AWGN) channel, as well as to various iterative decoding algorithms that fall into the category of message passing algorithms. The density involved in this numerical technique is the probability density function (pdf) of the messages passed at each decoding iteration. Under a tree-like assumption, density evolution computes the exact density of messages as they evolve from iteration to iteration, and can be used to evaluate the asymptotic bit-error probability of the code ensemble for each iteration. Allowing for infinite iterations, it can be used to determine the decoding threshold, which is the boundary of the error-free region for the asymp- totic performance of the code ensemble when the code length tends to infinity. Based on the performance concentration the- orem and cycle-free convergence theorem presented in [1], the threshold value can be understood as the bound on achievable performance of an actual LDPC code with a sufficiently large block length after a sufficient number of iterations. The design of a good LDPC code for a certain channel is accomplished by searching for a near-optimum degree distribution pair with the largest threshold [1]. Using density evolution and linear search techniques, Richardson et al. designed irregular LDPC codes capable of very closely approaching the Shannon limit at very large block lengths [1], [6]. For example, a carefully designed rate- irregular LDPC code with a block length of bits ( information bits) exhibits a bit-error rate (BER) of at just 0.0045 dB away from the Shannon limit in the AWGN channel [6]. Full density evolution for AWGN channels is computation- ally intensive. Chung et al. [2] proposed a simplified version to estimate the threshold with sum–product decoding of binary LDPC codes over AWGN. Their idea is to approximate the density of messages (expressed as log-likelihood ratio (LLR) values) by a Gaussian distribution. Using the symmetry property of the messages [1], the variance of Gaussianly distributed mes- sages is found to be equal to twice the mean [1]. The problem of calculating the message distribution over a one-dimensional (1-D) real space then collapses to the problem of updating a single number, i.e., the mean of the messages. Although binary LDPC codes are known to be powerful enough to achieve channel capacity, it is nevertheless worth exploring the potential of nonbinary codes, especially for channels with input from high-order constellations, which in principle support higher capacity than those employing binary 0018-9448/$25.00 © 2009 IEEE