# Hard-Decision Coded Modulation for High-Throughput Short-Reach Optical Interconnect
## 3 Performance Over AWGN Channel  
In this section, we show the performance over the most commonly used channel for optical communications, i.e., AWGN channel, which neglects the correlation caused by dispersion and nonlinear interference. Figure 2 shows a schematic diagram of the proposed CM communication system. The channel block in Figure 2 is modeled under a circularly symmetric Gaussian noise assumption, i.e., \(p_{\mathbf{Y}|\mathbf{X}}\) is fully determined by the variance \(\sigma_{z}^{2}\), which corresponds to the total noise power per one complex dimensions. By solving the optimization problem in Equation (3), we obtained three groups of geometrically-shaped constellations with SEs of 6 bit/2D-sym, 7 bit/2D-sym, and 8 bit/2D-sym as follows. * For the geometrically-shaped 64-point modulation format (64-GS), we optimized the constellation and its labeling for SNR = 19 dB. The 64-GS constellation and left-MSB (most significant bit) bits mapping are plotted in Figure 3. The 64-GS format with \(M=2^{m}=64\) points is labeled by \(m=6\) bits, which are represented by \((b_{1},b_{2},b_{3},b_{4},b_{5},b_{6})\) and the last two bits \((b_{5},b_{6})\) determine the quadrants of the constellation point (marked in different color). The first four bits determine the one of the 16 points in each quadrant. * For the geometrically-shaped 128-point modulation format (128-GS), we optimized the constellation and its labeling for SNR = 23 dB. The 128-GS constellation and bits mapping are plotted in Figure 4a. * For the geometrically-shaped 256-point modulation format (256-GS), we optimized the constellation and its labeling for SNR = 25 dB. The 256-GS constellation and bits mapping are plotted in Figure 4b. The 256-GS format with \(M=2^{m}=256\) points is labeled by \(m=8\) bits, which are represented by \((b_{1},b_{2},\ldots,b_{8})\), and the two bits \((b_{4},b_{8})\) determine the quadrants of the constellation point (marked in different color). The other six bits determine the one of 64 points in each quadrant. Figure 4: Constellation and labeling of 128-GS and 256-GS optimized for SNR = 23 dB and SNR = 25 dB, respectively. Figure 3: Constellation and labeling of 64-GS optimized for SNR = 19 dB with left-MSB bits mapping. The symbols in the same color are determined by the last two bits \(b_{5}\) and \(b_{6}\). For SCCs, two component codes, i.e., BCH codes \((504,485,2)\) and \((228,209,2)\), are considered. They correspond to the acceptable overhead range between 7% and 20% in optical fiber communications, which are the highest and lowest code rates investigated in Figure 1. These parameters are obtained by shortening the 1 bit-extended BCH code \((512,493,2)\) by 8 and 284 bits, respectively. The extended bit is a parity bit and added at the end of standard BCH code \((511,493,2)\). These two BCH codes result in SCC code rates \(R\approx 0.92\) and \(R\approx 0.83\), respectively, and SCC block size \(w=252\) and \(w=114\), respectively The decoding window size is 8, and the number of iterations is 7. The threshold \(\delta\) for marking HRBs is 5. Note that the threshold here is equivalent to \(\delta=10\) in [47] due to the different definition of the SNR. Here, the SNR is defined as the energy per complex symbol over the noise variance over one complex dimension. Figure 5 shows the post-FEC BER performance vs. SNR for SCC code rates of \(R=0.83\) (circles) and 0.92 (triangles). Furthermore, we investigate three CM schemes with \(M=\{64,128,256\}\): _M_QAM with SCC standard decoding (solid lines), _M_-GS with SCC standard decoding (dashed lines), and _M_-GS with SABM decoding (dotted lines). It can be seen from Figure 5 that for different code rates, 64-GS can provide up to 0.29 dB gain compared to 64 QAM. In addition, 64-GS with SABM decoding further increases the gain up to 0.55 dB at the BER of \(10^{-6}\). For \(M=128\) and \(M=256\), the gains increase up to 0.83 dB and 0.63 dB, respectively Note that for constellations representing an odd number of bits, e.g., 128 QAM, it is not possible to obtain a Gray mapping.