## Abstract

We propose a coded *N*-dimensional modulation scheme suitable for ultra-high-speed serial optical transport. The proposed scheme can be considered as a generalization of OFDM, and hence, we call it as generalized OFDM (GOFDM). In this scheme, the orthogonal subcarriers are used as basis functions and the signal constellation points are defined over this *N*-dimensional linear space. To facilitate implementation, we propose using *N*-dimensional pulse-amplitude modulation (ND-PAM) as the signal constellation diagram, which is obtained as the *N*-ary Cartesian product of one-dimensional PAM. In conventional OFDM, QAM/PSK signal constellation points are transmitted over orthogonal subcarriers and then they are multiplexed together in an OFDM stream. Individual subcarriers, therefore, carry *N* parallel QAM/PSK streams. In the proposed GOFDM scheme instead, an *N*-dimensional signal constellation point is transmitted over all *N* subcarriers simultaneously. When some of the subcarriers are severely affected by channel impairments, the constellation points carried by those subcarriers may be lost in the conventional OFDM. In comparison, under such conditions, the overall signal constellation point will face only small distortion in GOFDM and it can be recovered successfully using the information on the other high fidelity subcarriers. Furthermore, because the channel capacity is a logarithmic function of signal-to-noise ratio but a linear function of the number of dimensions, the spectral efficiency of optical transmission systems can be improved with GOFDM.

© 2011 OSA

## 1. Introduction

Optical communication systems have been rapidly evolving in recent years in order to meet continuously increasing demands on transmission capacity, originating mostly from the Internet and multimedia applications [1–4]. In order to satisfy high capacity demands, according to some industry experts, the 1 TbE standard should be completed by 2013 [5]. Coherent optical OFDM is one possible pathway towards achieving beyond 1 Tb/s optical transport [2–4]. Unfortunately, these initial studies indicate that the system Q-factor when multiband OFDM with orthogonal sub-bands is used is high, about 13.2 dB after 1000 km of SMF [2–4], which represents a very tight margin in terms of 7% overhead for the ITU-T standard RS(255,239) code. Under such stringent requirements, strong LDPC codes serve better than the standard RS(255,239) code as forward error correction (FEC) mechanisms [6–14]. Another approach is based on multidimensional coded modulation [7–9]. Namely, by increasing the number of dimensions, i.e., the number of orthonormal basis functions, we can increase the aggregate data rate of the system without degrading the bit error rate (BER) performance as long as the orthogonality among basis functions is preserved. Most of the papers on multidimensional coded modulation for optical communications so far have been related to single carrier systems. In this paper, we extend our multidimensional coded modulation approach [7–9] to include multicarrier systems.

The proposed coded multidimensional modulation scheme is called as generalized OFDM (GOFDM), which can also be considered as an *N*-dimensional pulse-amplitude modulation (PAM), and hence, it can also be called as *N*D-PAM. In GOFDM scheme, *N* orthogonal subcarriers are used as basis functions to define an *N*-dimensional linear function space. Although any *N*-dimensional signal constellation can be used in the proposed scheme, we choose to employ *N*D-PAM signal constellation, which is obtained by the *N*-ary Cartesian product of one-dimensional (1D) PAM. The benefit we gain from using *N*D-PAM as a signal constellation diagram instead of using a set of constellation points randomly chosen over the *N*-dimensional space is the simplicity in implementation due to the ease of generating 1D-PAM and its *N*-ary Cartesian product, i.e., *N*D-PAM. In conventional OFDM, QAM/PSK signal constellation points are transmitted over orthogonal subcarriers and then multiplexed together in an OFDM stream. Individual subcarriers therefore carry *N* parallel QAM/PSK streams. Thus, when some of the subcarriers are severely degraded, the information carrier over those subcarriers will most probably be unrecoverable. In contrast, in GOFDM, an *N*-dimensional signal constellation point is simultaneously transmitted over all *N* subcarriers, which serve as the individual basis functions as we mentioned above. Even if some of the subcarriers are severely affected by channel distortion, the *N*-dimensional signal constellation point will face only a partial distortion, which can be further reduced by using strong, channel-capacity-achieving channel codes. Furthermore, because channel capacity is a logarithmic function of signal-to-noise ratio (SNR) but a linear function of the number of dimensions (*N*), the spectral efficiency of optical transmission systems can be dramatically improved by proposed *N*-dimensional GOFDM scheme. Notice, however, that the complexity of symbol log-likelihood ratio (LLR) calculation operation increases with the number of dimensions, and it is clear that, in practice, three to seven dimensions should be used. As a workaround, we also propose a frequency-interleaved scheme that meticulously combines subsystems with reasonable number of (three to five) dimensions into a system with multi-Tb/s serial aggregate data rate.

The remainder of the paper is organized as follows. The proposed LDPC-coded GOFDM is described in Section 2. In Section 3, we describe the frequency-interleaved GOFDM as an enabling technology for multi-Tb/s serial optical transport. For completeness of the presentation, the quasi-cyclic binary/nonbinary LDPC codes are described in Section 4. Performance analysis and our discussion on their significance are provided in Section 5. Finally, some important concluding remarks are given in Section 6.

Notice that initial idea of using subcarriers as dimensions has been presented in our recent paper [17]. However, ref [17]. is related to multicarrier systems rather than OFDM. The complexity of such systems is much higher than that of GOFDM systems. Moreover, the nonbinary LDPC-code performance results have not been provided at all.

## 2. Description of proposed GOFDM system

The *N*D-PAM signal constellation is obtained by the *N*-ary Cartesian product of the 1D-PAM signal constellation. The 1D-PAM signal constellation is described by the following set of signal amplitudes $X=\left\{\left(2i-1-L\right)d,\text{}i=1,\text{}2,\text{}\dots ,\text{}L\right\},$ where 2*d* is the Euclidean distance between two neighboring signal amplitudes. The *N*D-PAM signal constellation is therefore obtained by

For example, for $L=4$ and $N=3,$ the corresponding constellation diagram in Eq. (1) is given by ${X}^{3}=X\times X\times X=\left\{\left({x}_{1},\text{}{x}_{2},\text{}{x}_{3}\right)|{x}_{i}\in X=\{-3,\text{}-1,\text{}1,\text{}3\},\text{\hspace{0.17em} \hspace{0.17em}}1\le i\le 3\right\}.$ Thus, the number of points in the *N*D-PAM constellation is given by $M={L}^{N},$ and hence, the number of bits transmitted per symbol when this constellation is used becomes $b={\mathrm{log}}_{2}\left({L}^{N}\right).$ For the same total bandwidth (of both *M*-QAM-OFDM and GOFDM) the spectral efficiency for GOFDM is ${\mathrm{log}}_{2}\left({L}^{N}\right)/{\mathrm{log}}_{2}\left(M\right)$ (*N*>2) times larger, which will become evident in Sec. 5, were several illustrative examples are given . In order to make a clear distinction between several *N*D-PAM schemes with the same number of dimensions *N*, we use *L ^{N}*-

*N*D-PAM notation, which unambiguously defines the signal constellation as the one with

*N*dimensions and with

*L*1D-PAM constellation points per dimension. For example, for $L=4$ and $N=3,$ the resulting signal constellation diagram assumes the name 4

^{3}-3D-PAM. Notice that square-QAM can also be described as 2D Cartesian product of PAM.

The transmitter configuration of a system using the proposed GOFDM scheme with *L ^{N}*-

*N*D-PAM signal constellation is depicted in Fig. 1(a) . As seen in the figure,

*b*independent binary data streams are encoded in parallel using

*b*identical binary $\text{LDPC(}n,k\text{)}$ codes of code rate $r=k/n,$ where

*n*and

*k*denote the codeword length and the information word length, respectively. (We should note here that different LDPC codes can be used to protect information streams on each one of the

*b*branches. Here, without loss of generality, we used identical LDPC code to simplify the notation.) The codewords are written row-wise into a $b\times n$ bit interleaver. At each symbol slot, the

*N*D-mapper reads a

*b*-bit symbol from the bit interleaver column-wise and outputs the constellation point corresponding to that symbol. The

*N*D-mapper is implemented as a look-up table (LUT) with

*b*input bits serving as a memory address that select the

*N*-coordinates of the corresponding ND-PAM signal constellation point. For example, the LUT for $L=4$ and $N=3$ (4

^{3}-3D-PAM) is shown in Table 1 . Upon mapping, the inverse fast Fourier transform (IFFT) is applied to perform modulation. We should stress that not all the subcarriers need to be used for modulation. In fact, we propose using

*N*out of

*N*

_{sc}available subcarriers $(N<{N}_{\text{sc}})$ as basis functions for the GOFDM scheme. The remaining subcarriers, i.e., $({N}_{\text{sc}}-N)$ of them, are employed in pilot estimation. After cyclic extension and parallel-to-serial (P/S) conversion, we perform digital-to-analog (DAC) conversion as shown in Fig. 1(a). Finally, the components corresponding to real and imaginary parts are used to modulate in-phase (I) and quadrature (Q) RF inputs of the I/Q-modulator on the corresponding polarization branch. The signals at the output of I/Q modulators located on the x- and y-polarization branches are then combined into single stream via polarization beam combiner (PBC) as shown in Fig. 1(a) to be coupled into the fiber.

The transmitter configuration above is described from the binary-LDPC-coded transmission perspective. If nonbinary LDPC codes are used, the transmitter configuration can be greatly simplified. To be precise, *b* binary LDPC encoders and the interleaver (see shaded block in Fig. 1(a)) are replaced by a single nonbinary LDPC encoder that outputs *b*-bit symbols ready to be mapped to constellation points. Thus, the need for the bit-to-symbol interface is avoided. Starting with the *N*D-mapper block the same configuration applies to both binary- and nonbinary-LDPC-coded transmission scenarios.

The receiver configuration is presented in Fig. 1(b). Using the polarization beam splitter (PBS), the optical signal is split into two orthogonal polarizations that are used as inputs into two balanced coherent detectors. On a given polarization branch, the I and Q outputs of the balanced coherent detector are sampled and passed through analog-to-digital conversion (ADC) units. The resulting samples on the I and Q branches are then combined into a single complex sequence by treating the I-branch samples as the real part and Q-branch samples as the imaginary part of the complex sequence. Following cyclic extension removal and serial-to-parallel conversion, this complex sequence is passed through an FFT block in order to perform demodulation. The *N* outputs of the FFT block, corresponding to *N*-dimensions of the *N*D-PAM signal constellation, are used as inputs to an *a posteriori* probability (APP) demapper.

The APP demapper calculates the symbol log-likelihood ratios (LLRs) according to the following equation.

**r**denotes the received

*N*-tuple at a given symbol interval, ${s}^{(a)}$ denotes the

*N*-tuple (i.e.,

*N*coordinates of the constellation point) at the output of the

*N*D-mapper corresponding to the

*b*-bit symbol $a=({a}_{1},\text{}{a}_{2},\text{}\dots ,\text{}{a}_{b}),$ where ${a}_{i}\in \left\{0,\text{}1\right\},\text{}1\le i\le b,$ and where $0=(0,\text{}0,\text{}\dots ,\text{}0)$ denotes the

*b*-bit all-zero symbol. The term $P\left({s}^{(a)}|r\right)$ in Eq. (2) is determined by Bayes’ rule as

*a priori*probability of the symbol $a.$ In the quasi-linear regime commonly-encountered in practice, Gaussian approximation can be used to compute $P\left(r|{s}^{(a)}\right).$ Since the binary LDPC decoders operate on bit LLRs, the APP demapper’s symbol LLR estimates must be converted to bit LLRs on bits ${a}_{j},\text{}1\le j\le m,$ comprising the symbol $a=({a}_{1},\text{}{a}_{2},\text{}\dots ,\text{}{a}_{b}).$ The operation of the bit LLR calculator is governed by

*j*th binary LDPC decoder as the bit LLR estimate at the current symbol interval. As shown in Fig. 1, the bit LLRs are forwarded to LDPC decoders, which enhance the bit LLR estimates through decoding, and feed the extrinsic bit LLRs back to the demapper for the demapper to use them as prior estimates on $P({s}^{(a)}).$ Then the APP demapper provides the binary LDPC decoders extrinsic bit LLRs for another round of iteration. This iterative extrinsic information exchange between the APP demapper and binary LDPC decoders continues for a predefined number of iterations, after which the binary LDPC decoder outputs are taken as the final estimates on the information sequences sent by the transmitter.

Similar to the transmitter side, the use of nonbinary LDPC codes greatly simplifies the receiver configuration. To elaborate, when nonbinary LDPC codes are used, *b* binary LDPC decoders (see shaded block in Fig. 1(b)) and the bit LLR calculation block are replaced by a single nonbinary LDPC decoder (depicted below the shaded block in Fig. 1(b)). Moreover, the use of a nonbinary LDPC decoder eliminates the need for the iterative information exchange between the dempaping and the decoding blocks, which in addition to decreasing computational complexity, reduces the latency. Finally, nonbinary LDPC-coded modulation schemes provide much larger coding gains compared to their corresponding bit-interleaved LDPC-coded modulation schemes, as we have shown in [10,11].

The aggregate information bit rate of this scheme is 2*rbR*
_{s} bits per second, where *R*
_{s} is the symbol rate. For example, by setting $L=4,$
$N=4,$ and ${R}_{\text{s}}=31.25\text{Giga-Symbols/s(GS/s),}$ the aggregate information bit rate becomes 400 Gb/s, which is compatible with 400 Gb/s Ethernet (400 GbE). If we increase the number of dimensions to $N=10$ while keeping all the other parameters the same, the aggregate information bit rate reaches 1 Tb/s, which is compatible with 1 Tb/s Ethernet (1 TbE). We should note here that the symbol rate is dictated by commercially available electronics, and hence, we used ${R}_{\text{s}}=31.25\text{GS/s}$ in our calculations. As the technology advances and much faster and more capable devices become available, the operating symbol rate will increase, so will the aggregate information bit rate.

As the final remark of this section, we compare the proposed GOFDM scheme with the conventional OFDM scheme. GOFDM transmits a given signal constellation point over all subcarriers simultaneously as we described above. On the other hand, in OFDM different MPSK/QAM sequences are used on different subcarriers, and then these subcarriers are multiplexed together in a single OFDM symbol. Therefore, if a particular subcarrier is severely affected by channel impairments during long-haul transmission, the information symbol will be lost if conventional OFDM is employed. If the proposed scheme is used, however, only one particular coordinate will be affected, and the transmitted symbol can be still recovered using the information gathered from the other subcarriers. Consequently, the proposed scheme will be much more efficient in dealing with various channel impairments. In addition, the proposed GOFDM scheme employs *N*-dimensional signal constellations while in OFDM only 2D signal constellations are used. Since, for the same symbol energy, the Euclidean distance between signal constellation points is much larger in *N*-dimensional space $(N\ge 3)$ than in 2D space, GOFDM achieves a much better optical SNR (OSNR) sensitivity as shown later in performance analysis.

## 3. Description of frequency-interleaved GOFDM

Here we describe the conceptual idea behind frequency interleaving/deinterleaving to enable beyond multi-Tb/s Ethernet based on GOFDM. In theory, we can increase the aggregate data rate by simply increasing the number of subcarriers as long as the orhtogonality among subcarriers is preserved. However, the complexity of APP demaper (see Fig. 1(b)) increases with *N*. To keep the complexity of the APP demaper reasonably low, we can employ the following approach. We first split the total number of subcarriers ${N}_{\text{sc}}={N}^{2}$ into *N* subgroups of *N* subcarriers. Next, the *k*th group of subcarriers $(k=1,\text{}2,\text{}\mathrm{...}\text{},\text{}N)$ to be used in the *N*-dimensional signal constellation is formed by taking the *k*th element of all subgroups. Finally, we perform encoding, modulation, transmission, demodulation, decoding on all groups as shown in Fig. 1. In such a way, if several subcarriers (coordinates) are affected by channel distortion, they will belong to different constellation points. Hence, via frequency interleaving/deinterleaving the immunity of GOFDM to channel distortion will be even more enhanced compared to the conventional OFDM. By using sufficiently high dimensionality of signal constellations $(N\ge 3),$ the OSNR improvement advantage will still be preserved.

## 4. Binary and nonbinary quasi-cyclic LDPC codes

The parity-check matrix of binary, regular quasi-cyclic LDPC (QC-LDPC) codes can be represented by [6,12]

**I**is the $B\times B$ identity matrix (

*B*is a prime number), $P=[{p}_{ij}]$ is the $B\times B$ permutation matrix whose entries are all zeros except ${p}_{i,i+1}={p}_{B-1,0}=1,$ for all $i,\text{}0\le iB,$ and where

*r*and

*c*represent the number of block-rows and block-columns in Eq. (6), respectively. (A (

*w*

_{c},

*w*)-

_{r}*regular*LDPC(

*n*,

*k*) code is a linear block code whose parity-check matrix

**H**contains exactly

*w*

_{c}1s in each column and exactly ${w}_{r}={w}_{c}n/(n-k)$ 1s in each column, where ${w}_{c}\ll n-k.$) In order to ensure good performance, as we explained in detail in [6], the set of integers

*S*should be carefully chosen from the set $\{0,\text{}1,\text{}\dots \text{},\text{}B-1\}$ so that the cycles of short length in the corresponding Tanner (bipartite) graph representation of Eq. (6) are avoided. (The Tanner graph of an LDPC(

*n*,

*k*) code is drawn according to the following rule: check node

*c*is connected to variable node

*v*whenever the element

*h*in the parity-check matrix $H=[{h}_{ij}]$ is a 1.) The nonbinary parity-check matrix of a QC-LDPC code over GF(

_{cv}*q*), i.e., the Galois field of

*q*elements, can be obtained by properly assigning nonzero elements from GF(

*q*) to the 1s in the parity-check matrix of the corresponding binary QC-LDPC code, as we described in [13]. In addition to their appealing structural advantages, QC-LDPC codes, when designed meticulously, perform as well as corresponding random LDPC codes. The well-known sum-product algorithm (SPA) used for decoding binary LDPC codes was generalized to decoder

*q*-ary LDPC codes and was given the name

*q*-ary SPA (QSPA) by Davey and MacKay [14]. They also proposed an efficient implementation of QSPA based on the fast Fourier transform (FFT), which we refer to as FFT-QSPA. FFT-QSPA is particularly efficient over the nonbinary fields whose order is a power of 2 since the complex arithmetic due to FFT can be avoided. In this paper, we use nonbinary LDPC codes over the extension fields of the binary field, i.e.

*q*= 2

*, for some positive integer*

^{b}*b*, in order to benefit from this nice property. Further details on FFT-QSPA and a reduced complexity variant of it can be found in [10,11] and references therein.

## 5. Performance analysis

The probability of correct detection for GOFDM is given by

where ${P}_{\text{s}}^{\left(L-PAM\right)}$ is the error probability of*L*-ary 1D-PAM:

*N*D-PAM is then

*d*is the Euclidean distance between two neighboring signal amplitudes (the subscript

_{N}*N*is used to denote the number of dimensions). The corresponding expression for

*M*-QAM is given by $\left(2/3\right)\left(M-1\right){d}_{2}^{2}.$ For the same symbol energy, the same aggregate data rate [

*b*= log

_{2}

*M*= log

_{2}(

*L*)] and the same symbol rate we can establish the following connection between

^{N}*N*D-PAM Euclidean distance and that of QAM:

Clearly, *N*D-PAM Euclidean distance squared increases exponentially with number of dimensions (for *N*>2) compared to ${d}_{2}^{2}$.

By expressing ${d}_{N}^{2}$ from Eq. (10) as a function of ${E}_{\text{ave}},$ the symbol error probability in Eq. (9) becomes

By expressing symbol energy in terms of bit energy *E*
_{b} as follows ${E}_{av}={E}_{b}{\mathrm{log}}_{2}{L}^{N},$
*P*
_{s} can be evaluated against bit energy-to-power spectral density ratio *E*
_{b}/*N*
_{0}. In Fig. 2(a)
, we provide symbol error probabilities obtained by Eq. (11) and compare them against those obtained via Monte Carlo simulations. An excellent agreement is observed. We can see also that an increase in the number of dimensions results in small performance degradation as long as the orhtogonality among subcarriers is maintained. In order to be consistent with existent optical and digital communication literature [18,19] on horizontal axis we use OSNR per information bit so that different multilevel/multidimensional schemes can be compared.

As an illustration of the potential of the proposed scheme, we show in Fig. 2(b) the BER performance of the binary LDPC(16935, 13550)-coded GOFDM schemes for symbol rate of 31.25 GS/s. As we expect, an increase in the number of dimensions for a fixed *L* results in negligible BER performance degradation. The comparison of corresponding curves with $L=4$ and $L=8$indicates that in order to increase the aggregate data rate it would be better to increase the number of subcarriers rather than the 1D-PAM signal constellation size. The 4^{3}-3D-PAM-based-GOFDM outperforms corresponding 64-QAM-OFDM with an impressive 4.28 dB additional coding gain at the BER of 10^{−8}. The polarization-multiplexed (PolMUX) 4^{4}-3D-PAM performs just slightly worse than PolMux 16-QAM-OFDM, but provides an aggregate information bit rate of $2\times {\mathrm{log}}_{2}({4}^{4})\times 31.25\text{GS/s}\times 0.8=400\text{Gb/s}$ and as such it is compatible with the future 400 GbE. In comparison, the aggregate information bit rate of PolMux 16-QAM-OFDM is only 200 Gb/s. If we instead use the GOFDM scheme with $L=4$ and $N=10,$then the resulting aggregate information bit rate reaches 1Tb/s. Therefore, the proposed GOFDM scheme is a coded modulation scheme enabling both 400 Gb/s and 1 Tb/s Ethernet.

We further evaluate BER performance of the proposed coded GOFDM scheme when a component nonbinary LDPC(16935, 13548) code, which is designed based on the binary LDPC(16935, 13550) code, as described in Section 4, is used. The results of the simulations are shown in Fig. 3
. For GOFDM based on 4^{3}-3D-PAM, the nonbinary coded modulation scheme outperforms the binary scheme by 0.87 dB. On the other hand, for GOFDM based on 8^{3}-3D-PAM, the nonbinary coded modulation scheme outperforms the binary scheme by even a larger margin of 1.29 dB. What is also interesting to notice from Fig. 3 is that the proposed *N*D-PAM signal constellation performs close to the optimum signal constellation based on sphere-packing method [15,16]. Namely, the nonbinary LDPC-coded 4D-PAM performs only 0.33 dB worse than corresponding optimum signal constellation; however, *N*D-PAM is much easier to implement than the optimum signal constellation, which is comprised of real number amplitudes up to 12 digits after the decimal point [16].

## 6. Conclusion

As response to continuously increasing demands on transmission capacity, we proposed the LDPC-coded generalized OFDM. The proposed scheme employs *N*-dimensional Cartesian product of PAM, called ND-PAM, as the signal constellation. In this scheme, the orthogonal subcarriers are used as bases functions. Because in proposed GOFDM scheme the *N*-dimensional signal constellation point is transmitted over all *N* subcarriers, even if during long-haul transmission some of the subcarriers are severely affected by channel distortion, the overall signal constellation point will face only small distortion. In addition, because the channel capacity is a linear function of number of dimensions, the spectral efficiency of optical transmission systems can be improved with GOFDM. This scheme is the next generations, both 400 Gb/s and 1 Tb/s, Ethernet enabling technology. The binary LDPC-coded GOFDM scheme significantly (>4dB) outperforms corresponding conventional QAM-OFDM counterpart. The use of nonbinary LDPC codes provides the additional improvement of 1.29 dB for GOFDM based on 8^{3}-3D-PAM. Notice that this paper was concerned with theoretical OSNR limits and coding gain improvements for proposed scheme, to clearly identify high potentials of proposed GOFDM scheme in terms of OSNR sensitivity and spectral efficiency.

## Acknowledgments

This work was supported in part by the National Science Foundation (NSF) under Grants CCF-0952711 and ECCS-0725405, in part by NSF CIAN ERC Center for Integrated Access Network under grant EEC-0812072, and in part by NEC Labs.

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