# Hard-Decision Coded Modulation for High-Throughput Short-Reach Optical Interconnect
## 1 Introduction  
Modern optical fiber communication systems have been revolutionized by digital coherent technology via techniques such as pulse shaping, digital signal processing (DSP), and forward error correction (FEC) [1]. In order to support the Internet's exponential traffic growth and improve the transmission reach, the most natural way of achieving this goal is using high spectral efficiency (SE) modulation formats and powerful FEC techniques. The combination of modulation format and FEC is also known as coded modulation (CM) [2, 3]. The purpose of CM is to increase the number of information bits per channel use (or information rate). High-order modulation formats can boost the information rates; however, there is often a gap between theperformance of a practical CM scheme and the corresponding theoretical limit. The gap is due to the use of uniform signaling with equidistant signal constellation points and the suboptimality of pragmatic FEC schemes. In order to enhance the performance of CM systems and reduce the two losses above, advanced FEC decoding architectures [3; 4; 5; 6] and signal shaping [7; 8; 9; 10; 11] have received considerable attention in the optical communication literature. The most popular approach is to use a binary code mapped nonbinary constellation in a so-called bit-interleaved coded modulation (BICM) structure. This approach naturally leads to a separate optimization of the FEC and the modulation format. One of the approaches to enhance the performance of the CM system is to employ signal shaping, which mimics a capacity achieving distribution. Signal shaping can be broadly categorized into probabilistic shaping (PS) [8; 9; 12; 13; 14] and geometric shaping (GS) [15; 16; 17; 18]. By employing the same number of signal dimensions, PS has superior achievable information rates (AIRs) performance improvement over \(m\)-QAM and flex-rate adjustment by adjusting probability for a finite number of constellation points with respect to GS (Section 4.2, [3]). However, its performance advantage over GS comes at the cost of high computational complexity. Usually, long codewords are required to achieve good performance, and error bursts after deshaping are also a problem. The inherent serialized processing also makes it challenging for parallelized implementation. These issues limit the application of PS in high-speed low-latency short-haul optical links. In GS, the constellation points have the same probability of occurrence, but they are not equally-spaced in the corresponding geometric space. Initially, GS was designed without taking the mapping strategy into account. For instance, many GS constellations are obtained by sphere packing arguments [19], and some of those were then used in fiber optical communications, e.g., in [20; 21]. Other approaches include for example iterative polar modulation with a Gaussian distribution maximizing the mutual information (MI) [22; 23]. Recently, end-to-end learning to maximize MI has been investigated in [24], and also experimentally demonstrated in [25]. Mapping-independent optimizations do not work well for BICM [26]. For BICM, the (mapping-dependent) generalized mutual information (GMI) should be optimized instead of MI. The challenge is to jointly design the geometry of the constellation and the mapping strategy. This problem has attracted attention in both academia [17; 27; 28; 29; 30; 31] and industry [32; 33; 34]. FEC design is also key to improve pragmatic CM schemes performance. In general, FEC comes in two flavors: hard-decision (HD) and soft-decision (SD) FEC. HD-FEC decoders use binary representations of bits, whereas SD-FEC decoders use more accurate representation of the bits' reliabilities. This \"soft information\" is often represented using logarithmic likelihood ratios (LLRs). Modern SD-FEC codes such as low-density parity-check (LDPC) offer a signal-to-noise ratio (SNR) improvement of \(\sim\)1-2 dB compared with HD-FEC codes of the same rate.