# Hard-Decision Coded Modulation for High-Throughput Short-Reach Optical Interconnect
## 2 AIRs, CM, and Proposed Design
### Advanced CM Schemes  
#### Improved Modulation  
GS constellations are generated by optimizing a cost function under certain constraints and typically for a given channel. The cost functions could be symbol error probability [56], MI [17; 22; 24], GMI [17; 28], etc. Throughout this paper, we consider BICM, which is one of the most popular CM schemes due to its advantages of flexibility, simplicity, and parallelizability. The transmitted symbols \(\mathbf{X}\) symbols with two real dimensions drawn uniformly from a discrete constellation \(\mathcal{X}\) with cardinality \(M=2^{m}=|\mathcal{X}|\). The \(j\)th constellation point is denoted by \(\mathbf{s}_{j}=\begin{bmatrix}s_{j,1},s_{j,2}\end{bmatrix}\in\mathcal{R}^{2}\) with \(j=1,2,\ldots,M\). We use the \(M\times 2\) matrix \(\mathbb{S}=[\mathbf{s}_{1}^{T},\mathbf{s}_{2}^{T},\ldots,\mathbf{s}_{M}^{T}]^{T}\) to denote the 2D constellation. The \(j\)th constellation point \(\mathbf{s}_{j}\) is labeled by the length-\(m\) binary bit sequence \(\mathbf{b}_{j}=[b_{j,1},\ldots,b_{j,m}]\in\{0,1\}^{m}\). The binary labeling matrix is denoted by a \(M\times m\) matrix \(\mathbb{B}=[\mathbf{b}_{1}^{T},\mathbf{b}_{2}^{T},\ldots,\mathbf{b}_{M}^{T}]^{T}\), which contains all unique length-\(m\) binary sequences. The 2D constellation and its binary labeling are fully determined by the pair of matrices \(\{\mathbb{S},\mathbb{B}\}\).  
Without the loss of generality, we consider only one of the polarizations for the shaping optimization; however, we emphasize that all results in this paper is performed for 2D constellations on each polarization. Therefore, we can optimize the two-dimensional constellation coordinates and the corresponding labeling by maximizing the cost function. For BW decoders, the AIR-based optimization problem is to find a constellation \(\mathbb{S}^{*}\) and labeling \(\mathbb{B}^{*}\) for a given channel law \(f_{\mathbf{Y}|\mathbf{X}}\) and an energy constraint, i.e.,  
\[\{\mathbb{S}^{*},\mathbb{B}^{*}\}=\underset{\mathbb{S},\mathbb{B}:E[\|\mathbf{X} \|^{2}]\leq\sigma_{x}^{2}}{\operatorname{argmax}}G(\mathbb{S},\mathbb{B},f_{ \mathbf{Y}|\mathbf{X}}) \tag{3}\]  
where \(G(\mathbb{S},\mathbb{B},f_{\mathbf{Y}|\mathbf{X}})\) is equal to either \(I_{\text{SD}}\) for SD-FEC or \(I_{\text{HD}}\) for HD-FEC; \(\sigma_{x}^{2}\) represents the transmitted power; and \(\mathbb{S}^{*}\) and \(\mathbb{B}^{*}\) indicate the optimal constellation and labeling, respectively. In this paper, average power constraint is considered in the optimization. Besides that, peak-to-average power ratio constraint is commonly used in optical fiber communication to maximize the signal generation performance for the systems with limited resolution of digital-to-analog converters and power budget of the links.  
Note that the optimization problem is a single objective function (AIR maximization) with multiple parameters. Many optimization methods can be used to find the optimal solution. This includes for example pairwise optimization algorithm [16; 27; 57], genetic algorithms [58], and machine learning approaches [59; 60; 24; 61; 62]. It is well known that a constellation with Gray-mapping has the minimum loss between MI and GMI. However, due to the unavailability of Gray-mapping for most GS constellations, only jointly-optimized coordinates and labeling is capable of reducing the impact of non-Gray mapping constellation, and thus reduce the gap between GMI and MI.  
The main advantage of GS over PS is its low implementation complexity, as only the mapping and demapping blocks of the transponder need to be modified by giving a look-up table. However, the GS constellations may result in more levels in both in-phase and quadrature branches. Therefore,the digital-to-analog converter/analog-to-digital converter may be required to be implemented with higher resolutions. In order to reduce the practical requirements, different constraints in Equation (3), such as symmetry, constant number of phase level, or amplitude, can also be used.