This letter investigates bit-interleaved coded multiple beamforming (BICMB) with perfect coding in millimeter-wave (mm-wave) massive multiple-input multiple-output (MIMO) systems to achieve both maximum diversity gain and multiplexing gain. Using perfect coding with BICMB enables us to do this. We show that by using BICMB and perfect coding, the diversity gain becomes independent from the number of transmitted data streams and the number of antennas in each remote antenna unit (RAU) at the transmitter and the receiver. The assumption is that the perfect channel state information (CSI) is known at both the transmitter and the receiver and the number of antennas goes to infinity. This latter assumption can be relaxed by a large number of antennas in each RAU, similar to the case for all massive MIMO research. Simulation results show that when the perfect channel state information assumption is satisfied, the use of BICMB with perfect coding results in the diversity gain values predicted by the analysis.
We study quantization of log-likelihood ratios (LLR) in bit-interleaved coded modulation (BICM) systems in terms of an equivalent discrete channel. We propose to design the quantizer such that the quantizer outputs become equiprobable. We investigate semi-analytically and numerically the ergodic and outage capacity over single- and multiple-antenna channels for different quantizers. Finally, we show bit error rate simulations for BICM systems with LLR quantization using a rate 1/2 low-density parity-check code.
Caire, Taricco and Biglieri presented a detailed analysis of bit interleaved coded modulation, a simple and popular technique used to improve system performance, especially in the context of fading channels. They derived an upper bound to the probability of error, called the expurgated bound. In this correspondence, the proof of the expurgated bound is shown to be flawed. A new upper bound is also derived. It is not known whether the original expurgated bound is valid for the important special case of square QAM with Gray labeling, but the new bound is very close to, and slightly tighter than,…
In this paper, we present the diversity order analysis of bit-interleaved coded multiple beamforming (BICMB) combined with the constellation precoding scheme. Multiple beamforming is realized by singular value decomposition of the channel matrix which is assumed to be perfectly known to the transmitter as well as the receiver. Previously, BICMB is known to have a diversity order bound related with the product of the code rate and the number of parallel subchannels, losing the full diversity order in some cases. In this paper, we show that BICMB combined with the constellation precoder and maxi…
Multiple beamforming is realized by singular value decomposition of the channel matrix which is assumed to be known to both the transmitter and the receiver. Bit-Interleaved Coded Multiple Beamforming (BICMB) can achieve full diversity as long as the code rate Rc and the number of employed subchannels S satisfy the condition RcS<=1. Bit-Interleaved Coded Multiple Beamforming with Constellation Precoding (BICMB-CP), on the other hand, can achieve full diversity without the condition RcS<=1. However, the decoding complexity of BICMB-CP is much higher than BICMB. In this paper, a reduced complexi…
The wideband regime of bit-interleaved coded modulation (BICM) in Gaussian channels is studied. The Taylor expansion of the coded modulation capacity for generic signal constellations at low signal-to-noise ratio (SNR) is derived and used to determine the corresponding expansion for the BICM capacity. Simple formulas for the minimum energy per bit and the wideband slope are given. BICM is found to be suboptimal in the sense that its minimum energy per bit can be larger than the corresponding value for coded modulation schemes. The minimum energy per bit using standard Gray mapping on M-PAM or …
Delayed bit-interleaved coded modulation (DBICM) generalizes bit-interleaved coded modulation (BICM) by modulating differently delayed sub-blocks of codewords onto the same signals. DBICM improves transmission reliability over BICM due to its capability of detecting undelayed sub-blocks with the extrinsic information of the decoded delayed sub-blocks. In this work, we propose a novel windowed decoding algorithm for DBICM, which uses the extrinsic information of both the decoded delayed and undelayed sub-blocks, to improve the detection on all sub-blocks. Numerical results show that the propose…
This paper investigates the design and performance of delayed bit-interleaved coded modulation (DBICM) with low-density parity-check (LDPC) codes. For Gray labeled square $M$-ary quadrature amplitude modulation (QAM) constellations, we investigate the optimal delay scheme with the largest spectrum efficiency of DBICM for a fixed maximum number of delayed time slots and a given signal-to-noise ratio. When analyzing the capacity of DBICM, we find two important properties: the capacity improvement due to delayed coded bits being mapped to the real and imaginary parts of the transmitted symbols ar…
A new variant of bit interleaved coded modulation (BICM) is proposed. In the new scheme, called Parallel BICM, L identical binary codes are used in parallel using a mapper, a newly proposed finite-length interleaver and a binary dither signal. As opposed to previous approaches, the scheme does not rely on any assumptions of an ideal, infinite-length interleaver. Over a memoryless channel, the new scheme is proven to be equivalent to a binary memoryless channel. Therefore the scheme enables one to easily design coded modulation schemes using a simple binary code that was designed for that binar…
This paper addresses the performance of bit-interleaved coded multiple beamforming (BICMB) [1], [2] with imperfect knowledge of beamforming vectors. Most studies for limited-rate channel state information at the transmitter (CSIT) assume that the precoding matrix has an invariance property under an arbitrary unitary transform. In BICMB, this property does not hold. On the other hand, the optimum precoder and detector for BICMB are invariant under a diagonal unitary transform. In order to design a limited-rate CSIT system for BICMB, we propose a new distortion measure optimum under this invaria…
It is shown that while the mutual information curves for coded modulation (CM) and bit interleaved coded modulation (BICM) overlap in the case of a single input single output channel, the same is not true in multiple input multiple output (MIMO) channels. A method for mitigating fading in the presence of multiple transmit antennas, named coordinate interleaving (CI), is presented as a generalization of component interleaving for a single transmit antenna. The extent of any advantages of CI over BICM, relative to CM, is analyzed from a mutual information perspective; the analysis is based on an…
The space-time bit-interleaved coded modulation (ST-BICM) is an efficient technique to obtain high diversity and coding gain on a block-fading MIMO channel. Its maximum-likelihood (ML) performance is computed under ideal interleaving conditions, which enables a global optimization taking into account channel coding. Thanks to a diversity upperbound derived from the Singleton bound, an appropriate choice of the time dimension of the space-time coding is possible, which maximizes diversity while minimizing complexity. Based on the analysis, an optimized interleaver and a set of linear precoders,…
When the Channel State Information (CSI) is known by both the transmitter and the receiver, beamforming techniques employing Singular Value Decomposition (SVD) are commonly used in Multiple-Input Multiple-Output (MIMO) systems. Without channel coding, there is a trade-off between full diversity and full multiplexing. When channel coding is added, both of them can be achieved as long as the code rate Rc and the number of employed subchannels S satisfy the condition RcS<=1. By adding a properly designed constellation precoder, both full diversity and full multiplexing can be achieved for both un…
For broadband wireless communication systems, Orthogonal Frequency Division Multiplexing (OFDM) has been combined with Multi-Input Multi-Output (MIMO) techniques. Bit-Interleaved Coded Multiple Beamforming (BICMB) can achieve both spatial diversity and spatial multiplexing for flat fading MIMO channels. For frequency selective fading MIMO channels, BICMB with OFDM (BICMB-OFDM) can be applied to achieve both spatial diversity and multipath diversity, making it an important technique. However, analyzing the diversity of BICMB-OFDM is a challenging problem. In this paper, the diversity analysis o…
This paper investigates the design of spatially coupled low-density parity-check (SC-LDPC) codes constructed from connected-chain ensembles for bit-interleaved coded modulation (BICM) schemes. For short coupling lengths, connecting multiple SC-LDPC chains can improve decoding performance over single-chains and impose structured unequal error protection (UEP). A joint design of connected-chain ensembles and bit mapping to further exploit the UEP from codes and high-order modulations is proposed. Numerical results demonstrate the superiority of the proposed design over existing connected-chain e…
This paper develops a low-complexity near-optimal non-coherent receiver for a multi-level energy-based coded modulation system. Inspired by the turbo processing principle, we incorporate the fundamentals of bit-interleaved coded modulation with iterative decoding (BICM-ID) into the proposed receiver design. The resulting system is called bit-interleaved coded energy-based modulation with iterative decoding (BICEM-ID) and its error performance is analytically studied. Specifically, we derive upper bounds on the average pairwise error probability (PEP) of the non-coherent BICEM-ID system in the …
In this paper, diversity analysis of bit-interleaved coded multiple beamforming (BICMB) is extended to the case of general spatial interleavers, removing a condition on their previously known design criteria and quantifying the resulting diversity order. The diversity order is determined by a parameter Qmax which is inherited from the convolutional code and the spatial de-multiplexer used in BICMB. We introduce a method to find this parameter by employing a transfer function approach as in finding the weight spectrum of a convolutional code. By using this method, several Qmax values are shown …
We revisit the information-theoretic analysis of bit-interleaved coded modulation (BICM) by modeling the BICM decoder as a mismatched decoder. The mismatched decoding model is well-defined for finite, yet arbitrary, block lengths, and naturally captures the channel memory among the bits belonging to the same symbol. We give two independent proofs of the achievability of the BICM capacity calculated by Caire et al. where BICM was modeled as a set of independent parallel binary-input channels whose output is the bitwise log-likelihood ratio. Our first achievability proof uses typical sequences, …
Bit-interleaved coded modulation (BICM) has attracted considerable attention from the research community in the past three decades, because it can achieve desirable error performance with relatively low implementation complexity for a large number of communication and storage systems. By exploiting the iterative demapping and decoding (ID), the BICM is able to approach capacity limits of coded modulation over various channels. In recent years, protograph low-density parity-check (PLDPC) codes and their spatially-coupled (SC) variants have emerged to be a pragmatic forward-error-correction (FEC…
An energy-efficient approach is presented for shaping a bit-interleaved low-density parity-check (LDPC) coded amplitude phase-shift keying (APSK) system. A subset of the interleaved bits output by a binary LDPC encoder are passed through a nonlinear shaping encoder whose output is more likely to be a zero than a one. The "shaping" bits are used to select from among a plurality of subconstellations, while the unshaped bits are used to select the symbol within the subconstellation. Because the shaping bits are biased, symbols from lower-energy subconstellations are selected more frequently than …
Device-to-device (D2D) aided beamforming for coded caching is considered in finite signal-to-noise ratio regime. A novel beamforming scheme is proposed where the local cache content exchange among nearby users is exploited. The transmission is split into two phases: local D2D content exchange and downlink transmission. In the D2D phase, users can autonomously share content with the adjacent users. The downlink phase utilizes multicast beamforming to simultaneously serve all users to fulfill the remaining content requests. We first explain the main procedure via two simple examples and then pre…
The capacity of the AWGN broadcast channel is achieved by superposition coding, but superposition of individual coded modulations expands the modulation alphabet and distorts its configuration. Coded modulation over a broadcast channel subject to a specific channel-input modulation constraint remains an important open problem. Some progress has been made in the related area of unequal-error protection modulations which can be considered single-user broadcast transmission, but it does not approach all points on the boundary of the capacity region. This paper studies broadcast coded modulation u…
In this paper we carry out the asymptotic diversity analysis for millimeter-wave (mmWave) massive multiple-input multiple-output (MIMO) systems by using bit interleaved coded multiple beamforming (BICMB). First, a single-user mmWave system which employs $M_t$ antenna subarrays at the transmitter and $M_r$ antenna subarrays at the receiver is studied. Each antenna subarray in the transmitter and the receiver has $N_r$ and $N_t$ antennas, respectively. We establish a theorem for the diversity gain when the number of antennas in each remote antenna unit (RAU) goes to infinity. Based on the theore…
Multi-Input Multi-Output (MIMO) techniques have been incorporated with Orthogonal Frequency Division Multiplexing (OFDM) for broadband wireless communication systems. Bit-Interleaved Coded Multiple Beamforming (BICMB) can achieve both spatial diversity and spatial multiplexing for flat fading MIMO channels. For frequency selective fading MIMO channels, BICMB with OFDM (BICMB-OFDM) can be employed to provide both spatial diversity and multipath diversity, making it an important technique. In our previous work, the subcarrier grouping technique was applied to combat the negative effect of subcar…
The Golden Code is a full-rate full-diversity space-time code, which achieves maximum coding gain for Multiple-Input Multiple-Output (MIMO) systems with two transmit and two receive antennas. Since four information symbols taken from an M-QAM constellation are selected to construct one Golden Code codeword, a maximum likelihood decoder using sphere decoding has the worst-case complexity of O(M^4), when the Channel State Information (CSI) is available at the receiver. Previously, this worst-case complexity was reduced to O(M^(2.5)) without performance degradation. When the CSI is known by the t…
An analysis of the diversity gain for bit-interleaved coded multiple beamforming (BICMB) method in millimeter-wave (mm-Wave) massive multiple-input multiple-output (MIMO) systems is carried out for both the single-user and multi-user scenario. We show that the diversity gain is independent of the number of data streams and full spatial multiplexing order can be achieved in both scenarios. Also, we show that the diversity gain in the multi-user scenario is independent of the number of users in the system and only depends on the number of the remote antenna units (RAUs) at the transmitter side, …
$2^m$-ary modulation creates $m$ bit channels which are neither independent nor identical, and this causes problems when applying polar coding because polar codes are designed for independent identical channels. Different from the existing multi-level coding (MLC) and bit-interleaved coded modulation (BICM) schemes, this paper provides a convolutional polar coded modulation (CPCM) method that preserves the low-complexity nature of BICM while offering improved spectral efficiency. Numerical results are given to show the good performance of the proposed method.
A framework is proposed that allows for a joint description and optimization of both binary polar coding and $2^m$-ary digital pulse-amplitude modulation (PAM) schemes such as multilevel coding (MLC) and bit-interleaved coded modulation (BICM). The conceptual equivalence of polar coding and multilevel coding is pointed out in detail. Based on a novel characterization of the channel polarization phenomenon, rules for the optimal choice of the labeling in coded modulation schemes employing polar codes are developed. Simulation results regarding the error performance of the proposed schemes on th…
We consider the joint design of polar coding and higher-order modulation schemes for ever increased spectral efficiency. The close connection between the polar code construction and the multi-level coding approach is described in detail. Relations between different modulation schemes such as bit-interleaved coded modulation (BICM) and multi-level coding (MLC) in case of polar-coded modulation as well as the influence of the applied labeling rule and the selection of frozen channels are demonstrated.
Perfect Space-Time Block Codes (PSTBCs) achieve full diversity, full rate, nonvanishing constant minimum determinant, uniform average transmitted energy per antenna, and good shaping. However, the high decoding complexity is a critical issue for practice. When the Channel State Information (CSI) is available at both the transmitter and the receiver, Singular Value Decomposition (SVD) is commonly applied for a Multiple-Input Multiple-Output (MIMO) system to enhance the throughput or the performance. In this paper, two novel techniques, Perfect Coded Multiple Beamforming (PCMB) and Bit-Interleav…
Two decoder structures for coded modulation over the Gaussian and flat fading channels are studied: the maximum likelihood symbol-wise decoder, and the (suboptimal) bit-wise decoder based on the bit-interleaved coded modulation paradigm. We consider a 16-ary quadrature amplitude constellation labeled by a Gray labeling. It is shown that the asymptotic loss in terms of pairwise error probability, for any two codewords caused by the bit-wise decoder, is bounded by 1.25 dB. The analysis also shows that for the Gaussian channel the asymptotic loss is zero for a wide range of linear codes, includin…
The performance of bit-interleaved coded modulation (BICM) with bit shaping (i.e., non-equiprobable bit probabilities in the underlying binary code) is studied. For the Gaussian channel, the rates achievable with BICM and bit shaping are practically identical to those of coded modulation or multilevel coding. This identity holds for the whole range of values of signal-to-noise ratio. Moreover, the random coding error exponent of BICM significantly exceeds that of multilevel coding and is very close to that of coded modulation.
Beamforming techniques that employ Singular Value Decomposition (SVD) are commonly used in Multi-Input Multi-Output (MIMO) wireless communication systems. In the absence of channel coding, when a single symbol is transmitted, these systems achieve the full diversity order provided by the channel; whereas when multiple symbols are simultaneously transmitted, this property is lost. When channel coding is employed, full diversity order can be achieved. For example, when Bit-Interleaved Coded Modulation (BICM) is combined with this technique, full diversity order of NM in an MxN MIMO channel trans…
Although BICM can significantly improves the BER performance by iteration processing between the demapping and the decoding in a traditional receiver, its design and performance in PNC system has fewer studied. This paper investigates a bit interleaved coded modulation (BICM) scheme in a Gaussian two-way relay channel operated with physical layer network coding (PNC). In particular, we first present an iterative demapping and decoding framework specially designed for PNC. After that, we compare different constellation mapping schemes in this framework, with the convergence analysis by using th…
In this paper, a combination of Space Time Block Coded Spatial Modulation with Hybrid Analog-Digital Beamforming (STBC-SM-HBF) for Millimeter-wave (mmWave) communications is proposed in order to take advantage of the merits of Spatial Modulation (SM), Space-Time Block Codes (STBC), Analog Beamforming (ABF), and digital precoding techniques while avoiding their drawbacks. This proposed system benefits from the multiplexing gain of SM, from the transmit diversity gain of STBC, and from the Signal-to-Noise Ratio (SNR) gain of the beamformer. The simulation results demonstrate that the Zero Forcin…
Multilevel coding (MLC) is a coded modulation technique which can achieve excellent performance over a range of communication channels. Polar codes have been shown to be quite compatible with communication systems using MLC, as the rate allocation of the component polar codes follows the natural polarization inherent in polar codes. MLC based techniques have not yet been studied in systems that use spatial modulation (SM). SM makes the polar code design difficult as the spatial bits actually select a channel index for transmission. To solve this problem, we propose a Monte Carlo based evaluati…
Bit-interleaved coded modulation (BICM) and multilevel coded modulation (MLC) are commonly used to combine polar codes with high order modulation. While BICM benefits from simple design and the separation of coding and modulation, MLC shows better performance under successive-cancellation decoding. In this paper we propose a hybrid polar coded modulation scheme that lies between BICM and MLC, wherein a fraction of bits are assigned to set-partition (SP) labeling and the remaining bits are assigned for Gray labeling. The SP labeled bits undergo sequential demodulation, using iterative demodulat…
We study downlink beamforming in a single-cell network with a multi-antenna base station serving cache-enabled users. Assuming a library of files with a common rate, we formulate the minimum transmit power with proactive caching and coded delivery as a non-convex optimization problem. While this multiple multicast problem can be efficiently solved by successive convex approximation (SCA), the complexity of the problem grows exponentially with the number of subfiles delivered to each user in each time slot, which itself grows exponentially with the number of users. We introduce a low-complexity…
We consider the problem of using polar codes with higher order modulation over AWGN channels. Unlike prior work, we focus on using modulation independent polar codes. That is, the polar codes are not re-designed based on the modulation used. Instead, we propose bit-permuted coded modulation (BPCM): a technique for using the multilevel coding (MLC) approach for an arbitrary polar code. The BPCM technique exploits a natural connection between MLC and polar codes. It involves applying bit permutations prior to mapping the polar code to a higher order modulation. The bit permutations are designed,…
We consider the special case of index coding over the Gaussian broadcast channel where each receiver has prior knowledge of a subset of messages at the transmitter and demands all the messages from the source. We propose a concatenated coding scheme for this problem, using an index code for the Gaussian channel as an inner code/modulation to exploit side information at the receivers, and an outer code to attain coding gain against the channel noise. We derive the capacity region of this scheme by viewing the resulting channel as a multiple-access channel with many receivers, and relate it to t…
A protograph-based low-density parity-check (LDPC) code design technique for bandwidth-efficient coded modulation is presented. The approach jointly optimizes the LDPC code node degrees and the mapping of the coded bits to the bit-interleaved coded modulation (BICM) bit-channels. For BICM with uniform input and for BICM with probabilistic shaping, binary-input symmetric-output surrogate channels are constructed and used for code design. The constructed codes perform as good as multi-edge type codes of Zhang and Kschischang (2013). For 64-ASK with probabilistic shaping, a blocklength 64800 code…
This paper investigates the bit-interleaved coded generalized spatial modulation (BICGSM) with iterative decoding (BICGSM-ID) for multiple-input multiple-output (MIMO) visible light communications (VLC). In the BICGSM-ID scheme, the information bits conveyed by the signal-domain (SiD) symbols and the spatial-domain (SpD) light emitting diode (LED)-index patterns are coded by a protograph low-density parity-check (P-LDPC) code. Specifically, we propose a signal-domain symbol expanding and re-allocating (SSER) method for constructing a type of novel generalized spatial modulation (GSM) constella…
In traditional cache-enabled small-cell networks (SCNs), a user can suffer strong interference due to contentcentric base station association. This may degenerate the advantage of collaborative content caching among multiple small base stations (SBSs), including probabilistic caching and coded caching. In this work, we tackle this issue by deploying multiple antennas at each SBS for interference management. Two types of beamforming are considered. One is matched-filter (MF) to strengthen the effective channel gain of the desired signal, and the other is zero-forcing (ZF) to cancel interference…
In this paper, the problem of joint oscillator phase noise (PHN) estimation and data detection for multi-input multi-output (MIMO) systems using bit-interleaved coded modulation (BICM) is analyzed. A new MIMO receiver that iterates between the estimator and the detector, based on the expectation-maximization (EM) framework, is proposed. It is shown that at high signal-to-noise ratios, a maximum a posteriori estimator (MAP) can be used to carry out the maximization step of the EM algorithm. Moreover, to reduce the computational complexity of the proposed EM algorithm, a soft decision-directed e…
In this paper, the impact of Forward Error Correction (FEC) code namely Trellis code with interleaver on the performance of wavelet based MC-CDMA wireless communication system with the implementation of Alamouti antenna diversity scheme has been investigated in terms of Bit Error Rate (BER) as a function of Signal-to-Noise Ratio (SNR) per bit. Simulation of the system under proposed study has been done in M-ary modulation schemes (MPSK, MQAM and DPSK) over AWGN and Rayleigh fading channel incorporating Walsh Hadamard code as orthogonal spreading code to discriminate the message signal for indi…
Multi-antenna coded caching combines a global caching gain, proportional to the total cache size in the network, with an additional spatial multiplexing gain that stems from multiple transmitting antennas. However, classic centralized coded caching schemes are not suitable for dynamic networks as they require prior knowledge of the number of users to indicate what data should be cached at each user during the placement phase. On the other hand, fully decentralized schemes provide comparable gains to their centralized counterparts only when the number of users is very large. In this paper, we p…
A protograph-based low-density parity-check (LDPC) code design technique for bandwidth-efficient coded modulation is presented. The approach jointly optimizes the LDPC code node degrees and the mapping of the coded bits to the bit-interleaved coded modulation (BICM) bit-channels. For BICM with uniform input and for BICM with probabilistic shaping, binary-input symmetric-output surrogate channels for the code design are used. The constructed codes for uniform inputs perform as good as the multi-edge type codes of Zhang and Kschischang (2013). For 8-ASK and 64-ASK with probabilistic shaping, cod…
Multiple antenna systems have been extensively used by standards designing multi-gigabit communication systems operating in bandwidth of several GHz. In this paper, we study the use of transmitter (Tx) beamforming techniques to improve the performance of a MIMO system with a low precision ADC. We motivate an approach to use eigenmode transmit beamforming (which imposes a diagonal structure in the complete MIMO system) and use an eigenmode power allocation which minimizes the uncoded BER of the finite precision system. Although we cannot guarantee optimality of this approach, we observe that ev…
A device-to-device (D2D) aided multi-antenna coded caching scheme is proposed to improve the average delivery rate and reduce the downlink (DL) beamforming complexity.} Novel beamforming and resource allocation schemes are proposed where local data exchange among nearby users is exploited. The transmission is split into two phases: local D2D content exchange and DL transmission. In the D2D phase, subsets of users are selected to share content with the adjacent users directly. {In this regard, a low complexity D2D mode selection algorithm is proposed to find the appropriate set of users for the…
The recent introduction of vector coded caching has revealed that multi-rank transmissions in the presence of receiver-side cache content can dramatically ameliorate the file-size bottleneck of coded caching and substantially boost performance in error-free wire-like channels. We here employ large-matrix analysis to explore the effect of vector coded caching in realistic wireless multi-antenna downlink systems. Our analysis answers a simple question: Under a fixed set of antenna and SNR resources, and a given downlink MISO system which can already enjoy an optimized exploitation of multiplexin…
Coded caching can be applied in wireless multi-antenna communications by multicast beamforming coded data chunks to carefully selected user groups and using the existing file fragments in user caches to decode the desired files at each user. However, the number of packets a file should be split into, known as subpacketization, grows exponentially with the network size. We provide a new scheme, which enables the level of subpacketization to be selected freely among a set of predefined values depending on basic network parameters such as antenna and user count. A simple efficiency index is also …
We analyze the achievable information rates (AIRs) for coded modulation schemes with QAM constellations with both bit-wise and symbol-wise decoders, corresponding to the case where a binary code is used in combination with a higher-order modulation using the bit-interleaved coded modulation (BICM) paradigm and to the case where a nonbinary code over a field matched to the constellation size is used, respectively. In particular, we consider hard decision decoding, which is the preferable option for fiber-optic communication systems where decoding complexity is a concern. Recently, Liga \emph{et…
