# Hard-Decision Coded Modulation for High-Throughput Short-Reach Optical Interconnect
## 2 AIRs, CM, and Proposed Design  
### Achievable Information Rates (AIRs)  
Achievable Information Rates (AIRs) have emerged as practical tools to design optical fiber communication systems, i.e., to design modulation formats and predict the performance of FEC [49; 50; 51]. The channel MI represents an upper limit on the AIRs for a given channel when a given modulation format is used along with an information-theoretic optimum receiver. However, the implementation of such a decoder is in general avoided for practical reasons. A widespread choice for performance prediction of (binary) SD-FEC is the GMI, which is suitable for bit-wise (BW) decoder (i.e., a standard BICM receiver) [3; 50]. Throughout this paper, random (row) vectors are denoted by \(\mathbf{X}\) and their corresponding realizations are denoted by \(\mathbf{x}\). Matrices are denoted by \(\mathbb{X}\). The transmitted bits are assumed to be independent and uniformly distributed, which implies uniform symbols \(\mathbf{X}\) (no PS is used). The receiver assumes a memoryless channel and also uses a bit-wise decoder (i.e., a standard BICM receiver). Under the assumptions above, the AIRs for SD-FEC and BW decoder is GMI  
\[I_{\text{SD}}=\sum_{i=1}^{m}I(B_{i};\mathbf{Y})=\sum_{i=1}^{m}\mathbb{E}\left[ \log_{2}\frac{f_{\mathbf{Y}|B_{i}}(\mathbf{Y}|B_{i})}{f_{\mathbf{Y}}(\mathbf{Y})}\right] \tag{1}\]  
where the bits mapped to the channel input \(\mathbf{X}\) are represented by the random variables \(\mathbf{B}=[B_{1},B_{2},\ldots,B_{m}]\), \(m=\log_{2}M\) is the number of bits per constellation point, \(\mathbf{Y}\) represents channel output, \(f_{\mathbf{Y}|B_{i}}(\mathbf{Y}|B_{i})\) is the channel law and \(I(B_{i};\mathbf{Y})\) is the MI between the bits and the symbols. As for HD-FEC schemes, we neglect the bit position asymmetry by assuming an ideal interleaver between BW demapper and HD-FEC decoder. Under the assumptions above, the AIRs are given by [52]  
\[I_{\text{HD}}=\sum_{i=1}^{m}I(B_{i};\hat{B}_{i})\triangleq m(1+p\log_{2}(p)+( 1-p)\log_{2}(1-p)) \tag{2}\]  
where \(\hat{B}_{i}\) represents the estimated bit after demapper and \(p\) corresponds to the average pre-FEC bit error rate (BER) across the \(m\) bit positions. Note that an average binary symmetric channel capacity might be a pessimistic AIR for generic HD-BW decoders. However, it is a more suitable AIR for HD-FEC decoders that disregard bit position asymmetry. Even though AIRs can be used to predict key metrics of communication systems, a gap between the AIR and the corresponding performance in practice is inevitable. In order to achieve a higher AIR, these penalties should be avoided. In the following, three SNR gaps, including the gap between HD-FEC and SD-FEC, shaping gap, and practical coding gap, are shown via an example. **Example 1**. : _AIRs in Equations (1) and (2) for 64 QAM are shown in Figure 1 (solid red and solid blue curves, respectively). These results show the theoretical gap (blue shaded area) of 1-2 dB. The SNR gap caused by using HD-FEC instead of SD-FEC is relatively small in the high-SNR regime (\(\approx\)1 dB) with respect to the low-SNR regime (\(\approx\)2 dB). Figure 1 also shows the gap (gray shaded area) between the Shannon capacity and ideal SD-FEC, which can be reduced by using a high-order and shaped modulation with an optimal distribution._  
_AIR analysis relies on an ideal assumption that there exists a FEC code with infinitely long codewords, which achieves an arbitrarily low BER at a code rate \(R=\text{AIR}\). However, codes with a finite block length and a complexity-limited implementation will lead to a loss with respect to the AIR prediction. Here, we show the performance of LDPC codes using soft-decision decoding and SCC codes using hard-decision decoding (without any soft information) to achieve a post-FEC BER below \(10^{-6}\). A simple concatenated outer code, i.e., BCH, can bring the post-SCC BER down to \(10^{-15}\).