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Near-Shannon-Limit Linear-Time-Encodable Nonbinary Irregular LDPC Codes Jie Huang, Shengli Zhou, and Peter Willett Dept. of Electrical and Computer Engineering, University of Connecticut, Storrs, CT 06269, USA

Abstract—In this paper, we present a novel method to construct nonbinary irregular LDPC codes whose parity check matrix has only column weights of 2 and t, where t ≥ 3. The constructed codes can be encoded in linear time and in a parallel fashion. Also, they can achieve near-Shannon-limit performance over both AWGN and Rayleigh fading channels with a moderate field size. Analysis based on nonbinary EXIT charts is presented. Codes constructed by the proposed method with block lengths ranging from 1, 000 bits to 10, 000 bits are simulated. Simulation results show that codes of rate 1/2 and length 10, 000 bits can achieve block-error-rate (BLER) of 10−5 within 1.1 dB and 1.3 dB away from the Shannon limits of the AWGN and Rayleigh channels, respectively. In an AWGN channel, codes of rate r = 8/9 and length around 4, 000 bits can achieve BLER of 10−5 within 1.1 dB from the Shannon limit, while codes of rate r = 15/16 and length around 9, 000 bits can achieve BLER of 10−5 within 1.0 dB from the Shannon limit. We conjecture that to further lower the error floor (e.g., to a BLER of 10−10 ), a column weight no less than 3 is preferred and may be necessary, especially for codes with high rate and over Galois fields of small to moderate sizes.

I. I NTRODUCTION Gallager’s binary low-density parity-check (LDPC) codes [1] are excellent error-correcting codes that achieve performance close to the benchmark predicted by the Shannon theory [2]. Davey and Mackay first investigated the extension of LDPC to a nonbinary Galois field GF(q) over the binaryinput additive-white-Gaussian-noise (AWGN) channel [3]. It was shown empirically that nonbinary LDPC can potentially have better performance than binary irregular LDPC codes [3]. LDPC codes over high order Galois field possess quite different properties from their binary counterparts. Operating on the binary field, LDPC cycle codes, whose parity check matrices have a fixed column weight of 2, have performance levelling off at very high block error rates (BLER). However, LDPC cycle codes over GF(q) can achieve near-Shannon-limit performance as q increases [4]. This is due to the fact that moving to a higher order field, the code’s minimum Hamming distance can dramatically increase and the code’s Hamming distance spectrum can be shaped to approach the optimal one [4]. Further, Monte Carlo simulations for codes having approximately infinite length by Davey and MacKay [5], [6] show that the column degree distribution of nonbinary LDPC codes should be very sparse for large q, approaching that of cycle codes. It has been shown that a special class of This work is supported by the Office of Navel Research through grants N00014-07-1-0429, N00014-07-1-0805 (YIP), and N00014-09-1-0704 (PECASE).

cycle codes — regular cycle codes, which have fixed column and row weights in their parity check matrices — are well structured and can be encoded in linear time and in a parallel fashion [7]. Therefore, the family of regular cycle codes are attractive when a large q is selected, say q ≥ 256. Cycle codes over small to moderate Galois fields (e.g., 4 ≤ q ≤ 64) suffer from performance loss due to a “tail” in the low weight regime of the distance spectrum [4]. However, a small to moderate q is desirable from both the decoding complexity and the codingmodulation size-matching perspectives [8]. In the context of underwater acoustic communication, we have proposed in [8] a code construction method that replaces an appropriate portion of columns in the parity check matrix of a cycle code by columns having weight t, where t > 2, to shape the code’s distance spectrum, i.e., increasing the code’s minimum distance and decreasing the multiplicities of low weight codewords. In other words, the proposed codes have mixed column weights of 2 and t in their parity check matrices. The motivation is as follows. Cycle codes over GF(q) can achieve the Shannon limit under maximum likelihood (ML) decoding as q increases [4]. Besides, having the least column weight, cycle codes over GF(q) perform the best that could be obtained with iterative decoding compared with the ML decoding. However their small minimum distances force the code away from the optimal performance under ML decoding for a small to moderate q. Increasing the code’s mean column weight (replacing part of columns of weight 2 by columns of weight t > 2) can increase the code’s minimum distance and decrease the multiplicities of low weight codewords simultaneously. Note that these codes can be encoded in linear time and in a parallel fashion, retaining the desirable features of a regular cycle code. The contributions of this paper are as follows. •

•

Ref. [8] contains only simulation results. Here we show that the proposed code construction can approach the Shannon capacity closely using the extrinsic information transfer (EXIT) chart analysis. Only rate 1/2 codes are simulated in Ref. [8] in certain settings. We here provide more simulation results for the rate 1/2 codes in Galois fields with different sizes. In addition, we show that the proposed method can construct high-rate codes that achieve near-Shannon-limit performance in both AWGN and Rayleigh fading channels, even with a small to moderate field size.

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II. T HE P ROPOSED C ODE C ONSTRUCTION Our approach is to replace a portion of weight-2 columns of the m × n parity check matrix H of a cycle code by columns of weight t > 2, (e.g., t = 3 or t = 4). Let n1 columns of H have weight 2 and n2 columns have weight t. The mean column weight is n2 2n1 + tn2 = 2 + (t − 2) (1) n n In order to achieve linear-time encodability, we restrict n1 ≥ m, that is, 0 ≤ n2 ≤ (n − m). Therefore, we have η=

2 ≤ η ≤ 2 + (t − 2)r

(2)

where r = (n − m)/n. The matrix H can be arranged as    (3) H = H1  H2 where H1 contains all weight-2 columns and H2 contains all weight-t columns. Clearly, H1 is of size m × n1 and H2 is of size m × n2 . Now we need to design H1 and H2 . To maximally benefit from the structure of regular cycle codes, we propose to use the following design rule. Note that H1 corresponds to the check matrix of a general cycle code. We would like H1 to be as close to a regular cycle code as possible. Specifically, we split the matrix as     (4) H = H1a  H1b  H2 where the matrix H1a is of size m × n1a and the matrix H1b is of size m × n1b . The number n1a is the largest integer not greater than n1 that can render d1a = 2nm1a an integer, that is, H1a is the largest sub-matrix of H1 that could be made d1a -regular meaning that it has fixed column weight of 2 and row weight of d1a . If n1a = n1 , then n1b = 0. If so, H1 itself can be made regular, which is a special case. The detailed design procedure is as follows. • Step 1: Specify the structure of H1a . Construct a cycle code of fixed row weight d1a using the design methodologies categorized in [9], including the progressive-edge-growth (PEG) algorithm [10]. • Step 2: Specify the structure of H1b and H2 . Apply the PEG algorithm to attach n1b columns of weight 2 and n2 columns of weight t to the matrix H1a . This way, the structure of H in (4) is established. • Step 3: Specify the non-zero entries of H1 . Note that the sub-matrix H1 = [H1a  H1b ] can be regarded as a parity check matrix of a cycle code. Hence, existing design criteria in the literature for cycle codes,

1 0.95 0.9 IE,VND/IA,CND

The paper is organized as follows. Section II specifies the proposed construction. Analysis based on nonbinary EXIT charts is presented in Section III. Simulation results for codes of rate-1/2 over AWGN and Rayleigh channels are reported in Section IV and V, respectively. Simulation results for high rate codes are collected in Section VI, while conclusions are drawn in Section VII.

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Fig. 1. Nonbinary EXIT charts of variable node decoding (VND) and check node decoding (CND). The mixed VND curve is lying above the swapped CND curve for successful iterative decoding. The operating SNR is 19.0 dB.

such as those presented in [9], can be applied to choose appropriate nonzero entries for H1 . • Step 4: Specify the non-zero entries of H2 . The nonzero entries of H2 are generated randomly with a uniform distribution over the set GF(q)\0. The proposed LDPC codes try to make a large portion of the check matrix come from a regular cycle code. This way, many benefits of a regular cycle code can be retained. In particular, the proposed nonbinary irregular LDPC code can be encoded in linear time and in a parallel fashion [8]. Further, special treatments on the nonzero entries for the sub-matrix H1 have been employed to improve the code’s error floor performance. III. N ONBINARY EXIT C HART A NALYSIS Reference [11] presented the nonbinary extrinsic information transform (EXIT) chart to analyze and design nonbinary irregular LDPC codes at infinite block length. In this section we will use these nonbinary EXIT charts to analyze the performance loss of the proposed method compared with the degree-distribution-optimized irregular nonbinary LDPC code. Take the first code presented in Section VII.F of reference [11] as an example where q = 32. A code of rate 0.6 is designed by setting the design SNR to be 18.5 dB and letting the column weight range from 2 to 80 (The obtained code has a maximum column weight of 30 though). 84.8% of the variable nodes have column weight of 2 in the obtained column weight distribution; see reference [11] for the details of the algorithm and its Matlab source. In comparison, we set the design SNR to be 19.0 dB and let the column weight range from 2 to 3, the algorithm will produce a code of rate 0.60089 where 90.9% of the variable nodes have column of weight 2 whereas 9.1% of the variable nodes have column of weight 3 in the obtained column weight distribution. Compared with the code from [11], the performance loss of the proposed construction is about 19.0 − 18.5 = 0.5 dB. We now set the design SNR to be 19.0 dB and let the column weight be chosen between 2 and

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

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Fig. 2. Performance comparison of irregular codes over GF(4), GF(8) and GF(16) with an optimized binary irregular LDPC code; r = 1/2 and the codeword length is 1008 bits.

Fig. 3. Performance comparison of irregular codes over GF(4), GF(8), GF(16) and GF(64) with an optimized binary irregular LDPC code; r = 1/2 and the codeword length is 2688 bits.

4, the algorithm of [11] will produce a code of rate 0.60045 where 95.5% of the variable nodes have column of weight 2 whereas 4.5% of the variable nodes have column of weight 4 in the obtained column weight distribution. Compared with the code from [11], the performance loss is also about 0.5 dB. The corresponding EXIT charts of our designed code with mixing column weights of 2 and 3 are shown in Fig. 1. It can be seen from Fig. 1 that the VND curve of degree 2 is close to the swapped CND curve while staying away from VND curves of degree larger than 2. This has been extensively observed in our EXIT chart based analyses for codes over GF(q) with q ≥ 32. When q gets smaller, the gap between the VND curve of degree 2 and VND curves of degree larger than 2 will shrink. However, the proposed construction can potentially still work very well, resulting in a column weight distribution with the percentage of nodes of degree larger than 2 increased. For example, the two degree-distribution-optimized nonbinary LDPC codes over GF(4) and GF(8) presented in Section VIII of [11] can be well approximated by codes with mixture degrees of 2 and t, where t is either 3 or 4. Note that the derived weight distribution from the nonbinary EXIT chart design is optimal in the sense of infinite length, without considering the error floor behavior of practical finite length codes. In reality, the code design has to consider the water-fall performance and the error floor performance. To achieve a target BLER, codes of different lengths could have different mean column weights, even though they share the same code rate. Thus the mean column weights of finite length codes can have deviations from the one obtained from the nonbinary EXIT chart design.

do we choose appropriate parameters for t, the mean column weight η as shown in (2), and the size of Galois field q? In contrast to binary LDPC codes, we have another degree of freedom to exploit, that is, the size of the Galois field q. The error floor of an LDPC code with iterative decoding depends on a number of structural properties of the code and its Tanner graph, such as girth, minimum Hamming distance, weight distribution of pseudo-codewords or near codewords [12], etc.. It is unknown how the error floor of an LDPC code is exactly affected by these factors. However, experimental results show that if a code has a large minimum Hamming distance and the minimum weight of pseudo-codewords is also large, then the code has a low error floor. The key concern for the proposed method is the minimum Hamming distance of the code. Other effects, such as low weight pseudo-codewords or near codewords, are not considered. Nevertheless, when we move to higher order Galois field and adopt irregular distributions, the detrimental effect of low weight pseudocodewords or near codewords can be alleviated significantly which has been verified in almost all simulations. For the proposed code construction, under given q, t and codeword length, for a given target BLER there exists an optimal value for the mean column weight η which is unfortunately unknown. Nevertheless, we can always increase η starting from 2.0 to alleviate the code’s error floor to meet the target BLER. The obtained η may have a deviation from the unknown optimal one which is expected to incur slight or negligible compromise of the code’s water-fall performance as shown in Section III with nonbinary EXIT chart analysis. In this and the next two sections, we perform simulations to evaluate the performance of the proposed codes, focusing on block-error-rate (BLER). In all simulations, the codewords are transmitted over the binary input AWGN or Rayleigh fading channels. For each SNR we run simulations until more than 20 block errors have been observed or up to 2, 000, 000 block decodings, so that a BLER down to 10−5 is obtainable. The

IV. S IMULATION R ESULTS OVER AWGN C HANNEL The fundamental question of the proposed code design is how to achieve both good water-fall performance and error-floor performance for different code rates and codeword lengths. Specifically, given a target BLER, say 10−5 , how

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

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Fig. 4. Performance comparison of irregular codes over GF(16) over Rayleigh channel; r = 1/2 and the codeword length is 2688 bits.

check matrices of all codes, including binary and nonbinary LDPC codes, are constructed by the progressive-edge-growth (PEG) method [10]. The PEG algorithm, which works in a best-effort greedy manner, can produce codes with Tanner graphs having a large girth [10]. The PEG algorithm has been shown to be efficient and feasible for constructing binary LDPC codes with short codeword lengths and high rates as well as binary LDPC codes with long codeword lengths [10]. For nonbinary LDPC codes, the decoder adopts the sequential belief propagation (BP) algorithm [13] with fast Fourier transform (FFT) based realization [14], and the maximum number of iterations is set to be 80. For binary LDPC codes, log domain sum-product decoder is applied with a maximum number of 80 iterations. Fig. 2 in [8] compared performance of the proposed LDPC codes over GF(16) having different mean column weights in an AWGN channel. All the codes there have rate of 1/2 and codeword length of 1008 bits; corresponding to 252 GF(16) symbols. Codes having mean column weight of 2.0, 2.2, 2.4, 2.6 and 2.8 were evaluated down to BLER of 10−5 . Two observations have been made, i) the performance curves of the codes with η = 2.0 and η = 2.2 level off at BLER above 10−5 due to the significant contribution from the probability of undetected errors. However, this is not the case if η ≥ 2.4; ii) As η increases from 2.4 to 2.6 and 2.8, the code performance degrades. The latter observation can be explained using nonbinary EXIT chart based analysis presented in section III. When η increases, say from 2.4 to 2.6, the threshold SNR also increases, incurring performance loss. Fig. 2 shows performance of the proposed nonbinary irregular LDPC codes over GF(4), GF(8) and GF(16) compared with a binary degree-distribution-optimized irregular code and MacKay’s regular-(3,6) code, all having rate of 1/2 and codeword length of 1008 bits (see more details on these codes’ parameters in [8]). It can be seen from Fig. 2 that the proposed method works well for an even small Galois field, say GF(4).

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Fig. 5. Performance comparison of the proposed LDPC codes with an degreedistribution-optimized binary LDPC code over Rayleigh channel; r = 1/2 and the codeword length is 2688 bits. The capacity limit is 1.830 dB.

Fig. 3 shows the performance of the proposed nonbinary irregular LDPC codes over GF(4), GF(8), GF(16) and GF(64) compared with binary degree-distribution-optimized irregular code, all having rate of 1/2 and codeword length of 2688 bits. It can be seen from Fig. 3 that all the proposed codes can achieve BLER of 10−5 within the Shannon limit by 1.3 to 1.6 dB. Compared with the regular cycle code over GF(64) with η = 2.0 and length of 1008 bits in Fig. 2, the regular cycle code over GF(64) with η = 2.0 and length of 2688 bits (not shown in Fig. 3) levels off at BLER above 10−5 . Larger mean column weight, say η = 2.2, is desired, as shown in Fig. 3. Compared with existing nonbinary LDPC codes in the literature, such as those in [15], the proposed codes achieve performance gains. In reference [15], a rate-1/2 128-ary (2032, 1016) quasi-cyclic LDPC code has been constructed, which has block length of 14224 bits and column weight of 3. This code achieves BLER of 10−5 at Eb /N0 of 2.0 dB. Our codes over GF(4) (GF(64), resp.), as shown in Fig. 3, achieve BLER of 10−5 at Eb /N0 of around 1.8 dB (1.6 dB, resp.), which is 0.2 dB (0.4 dB, resp.) better than the 128-ary (2032, 1016) quasi-cyclic LDPC code reported in [15], even though the block length of their code is more than 5 times longer than that of our codes. Besides, our codes have much less decoding complexity than the code in [15]. Further, the performance of codes over GF(16) and GF(64) with length of 10, 000 bits can approach the Shannon limit within 1.1 dB at BLER of 10−5 . V. S IMULATION R ESULTS OVER R AYLEIGH C HANNEL Fig. 4 compares performance of the proposed LDPC codes over GF(16) with different mean column weights in a Rayleigh fading channel. All the codes have rate of 1/2 and codeword length of 2688 bits; corresponding to 672 GF(16) symbols. Similar observations as for the AWGN case can be made. We observe from Fig. 4 that the performance curves of the codes

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

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with η = 2.0 and η = 2.2 level off at BLER above 10−5 due to the significant contribution from the probability of undetected errors. This is not the case if η ≥ 2.3. Actually no undetected errors have been observed for η ≥ 2.4 in our simulations (the curve for η = 2.4 is not shown in Fig. 4). Another interesting observation is that as η increases from 2.3 to 2.5 and 2.6, the code performance degrades. This can be explained using nonbinary EXIT chart based analysis. When η increases, say from 2.3 to 2.5, the threshold SNR also increases, incurring performance loss as shown in Fig. 4. We can say that the code with η = 2.3 is the best one for this particular example. Fig. 5 shows the performance comparison between the proposed nonbinary irregular LDPC codes and an optimized binary irregular LDPC code. The binary irregular code has a density-evolution-optimized degree distribution pair achieving an impressive iterative decoding threshold of 1.980 dB (the Shannon limit is 1.830 dB), from Table I in [16], i.e., the symbol-node edge distribution is 0.246544x + 0.230609x2 +

Fig. 9. Performance of nonbinary irregular LDPC codes of rate 15/16 and length around 9000 bits; The Shannon capacity for rate 15/16 is 3.9 dB.

0.002045x3 + 0.046487x5 + 0.150161x6 + 0.035344x7 + 0.004812x18 + 0.283998x19 and the check-node edge distribution is 0.000952x6 + 0.951871x7 + 0.047177x8 . It can be seen from Fig. 5 that all the proposed codes achieve similar or slightly better performance than the binary irregular code. Besides, all the codes can achieve BLER of 10−5 within the Shannon limit by 1.7 to 2.2 dB. By adopting an irregular column weight distribution, the code’s performance can be greatly improved, even for small to moderate q. Further, the performance of codes over GF(16) and GF(64) with length of 10, 000 bits can approach the Shannon limit of the Rayleigh fading channel within 1.3 dB at BLER of 10−5 . Besides, we can also seen from Figs. 3 and 5 that some irregular codes can perform very well over both AWGN and Rayleigh channels simultaneously, such as the GF(4) code with η = 2.8 and t = 4, and the GF(8) code with η = 2.7 and t = 4.

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.

VI. S IMULATION R ESULTS OF H IGH R ATE C ODES We now investigate the performance of the proposed nonbinary irregular LDPC codes with high rates, e.g., r = 8/9 and r = 15/16 over the binary input AWGN channel. Fig. 6 shows the simulation results of the proposed LDPC codes of rate r = 8/9 and length around 2000 bits. MacKay’s ‘s2.94.594’ code of rate-8/9 is also included in Fig. 6 [17]. Note that the cycle code over GF(64) having η = 2.0 starts to level off at BLER of 10−5 . The proposed irregular codes over GF(4), GF(8) and GF(16) start to show better performance than the cycle code over GF(64) with η = 2.0 at BLER of 10−5 . It can be seen from Fig. 6 that the proposed irregular LDPC codes achieve slightly better performance than MacKay’s code. Besides, all the proposed codes, including those over GF(4), can achieve BLER of 10−5 within the Shannon limit by 1.3 to 1.5 dB. Fig. 7 shows the simulation results of the proposed LDPC codes of rate r = 8/9 and length around 4000 bits which doubles that in Fig. 6. All the proposed codes, including those over GF(4), can achieve BLER of 10−5 within the Shannon limit by 0.9 to 1.1 dB. Furthermore, it can be seen from Fig. 6 and Fig. 7 that as the codeword length increases, higher mean column weight η is desired and even necessary. Fig. 8 shows the simulation results of the proposed LDPC codes of rate r = 15/16 and length around 4500 bits. MacKay’s ‘4376.282.4.9598’ code of rate-15/16 is also included in Fig. 8 [17]. Note that the cycle code over GF(64) having η = 2.0 starts to level off at BLER of 10−4 . The proposed irregular codes over GF(4), GF(8), and GF(16) start to show better performance than the cycle code over GF(64) with η = 2.0 at BLER of 10−4 . It can be seen from Fig. 8 that the proposed irregular LDPC codes achieve slightly better performance than MacKay’s code. Besides, all the proposed codes, including those over GF(4), can achieve BLER of 10−5 within the Shannon limit by 1.1 to 1.2 dB. Furthermore, it can be seen from Fig. 6 and Fig. 8 that as the code rate increases, larger mean column weight η is desired and even necessary. Fig. 9 shows the simulation results of the proposed LDPC codes of rate r = 15/16 and length around 9000 bits which doubles that in Fig. 8. All the proposed codes, including those over GF(4), can achieve BLER of 10−5 within the Shannon limit by 0.8 to 1.0 dB. Furthermore, it can be seen from Fig. 8 and Fig. 9 that as the codeword length increases, larger mean column weight η and larger t is desired and even necessary. Our experience with simulations is as follows. If the target BLER is 10−5 and the codeword length is less than 10000 bits, for large q (say q ≥ 256)), cycle codes with η = 2.0 are good enough, while for small to moderate q, higher η is desirable and even necessary. Setting t = 3 is usually good enough for moderate q (say 16 ≤ q ≤ 64) whereas for small q (say q ≤ 8) t = 4 can readily meet the goal. Further, we conjecture that to further lower the error floor (e.g., to a BLER of 10−10 ), a column weight no less than 3 is preferred and may be necessary, especially for codes with high rate and over Galois fields of small to moderate sizes.

VII. C ONCLUSION This paper presented a novel method to construct nonbinary irregular LDPC codes which can achieve near Shannon limit performance and can be encoded in linear time and in a parallel fashion. An analysis based on nonbinary EXIT chart was conducted. Codes of length from 1000 to 10000 bits and of high rates were evaluated. Further research can be suggested in several directions: i) evaluate the performance of the proposed nonbinary irregular LDPC codes down to extremely low BLER, say 10−10 or 10−15 , using hardware platform based emulations; ii) instead of using PEG for code construction, other techniques, such as graph conditioning, can be used to further lower the error floor for the proposed code with a fixed mean column weight; and iii) analyze the minimum distance and error floor properties of the proposed codes. R EFERENCES [1] R. G. Gallager, Low Density Parity Check Codes. Cambridge, MA: MIT Press, 1963. [2] T. Richardson, A. Shokrollahi, and R. Urbanke, “Design of capacityapproaching irregular low-density parity-check codes,” IEEE Transactions on Information Theory, vol. 47, no. 2, pp. 619–637, 2001. [3] M. C. Davey and D. MacKay, “Low-density parity-check codes over GF(q),” IEEE Communications Letters, vol. 2, pp. 165–167, 1999. [4] X.-Y. Hu and E. Eleftheriou, “Binary representation of cycle tannergraph GF(2b ) codes,” Proc. of International Conference on Communications, vol. 27, no. 1, pp. 528 – 532, June 2004. [5] M. C. Davey and D. MacKay, “Monte Carlo simulations of infinite low density parity check codes over GF(q),” in Proc. of Int. Workshop on Optimal Codes and related Topics, Bulgaria. [6] M. C. Davey, Error-Correction using Low-Density Parity-Check Codes. Dissertation, University of Cambridge, 1999. [7] J. Huang, S. Zhou, and P. Willett, “Structure of non-binary regular LDPC cycle codes,” in Proc. of ICASSP, Las Vegas, NV, Mar. 30-Apr. 4, 2008. [8] ——, “Nonbinary LDPC coding for multicarrier underwater acoustic communication,” IEEE J. Select. Areas Commun., vol. 26, no. 9, pp. 1684–1696, Dec. 2008. [9] ——, “Structure, property, and design of nonbinary regular cycle codes,” IEEE Transactions on Communications, submitted Oct. 2008; downloadable at http://www.engr.uconn.edu/˜jhuang/submittedTCOM.pdf. [10] X.-Y. Hu, E. Eleftheriou, and D.-M. Arnold, “Regular and irregular progressive edge-growth tanner graphs,” IEEE Transactions on Information Theory, vol. 51, no. 1, pp. 386–398, 2005. [11] A. Bennatan and D. Burshtein, “Design and analysis of nonbinary LDPC codes for arbitrary discrete-memoryless channels,” IEEE Transactions on Information Theory, vol. 52, no. 2, pp. 549–583, Feb. 2006. [12] T. Richardson, “Error floors of LDPC codes,” in Proc. of Allerton Conf. Commu., Control and Computing, Monticello, IL, Oct. 2003, pp. 1426– 1435. [13] H. Kfir and I. Kanter, “Parallel versus sequential updating for belief propagation decoding,” Physica A: Statistical Mechanics and its Applications, vol. 330, pp. 259–270, 2003. [14] H. Song and J. R. Cruz, “Reduced-complexity decoding of q-ary LDPC codes for magnetic recording,” IEEE Trans. Magn., vol. 39, pp. 1081– 1087, 2003. [15] B. Zhou, Y.-Y. Tai, L. Lan, S. Song, L. Zeng, and S. Lin, “Construction of high performance and efficiently encodable nonbinary quasi-cyclic LDPC codes,” in Proc. of Global Telecommunications Conference 2006, San Francisco, CA, Nov. 27-Dec. 1 2006, pp. 1–6. [16] J. Hou, P. H. Siegel, and L. B. Milstein, “Performance analysis and code optimization of low density parity-check codes on rayleigh fading channels,” IEEE J. Select. Areas Commun., vol. 19, no. 5, pp. 924–934, May 2001. [17] D. J. C. MacKay and E. A. Ratzer, “Gallager codes for high rate applications,” 2003, available from http://www.inference.phy.cam.ac.uk/mackay/.

978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.