Abstract-Codes on graphs of interest for next generation forward error correction (FEC) in high-speed optical networks, namely turbo codes and low-density parity-check (LDPC) codes, are described in this invited paper. We describe both binary and nonbinary LDPC codes, their design, and decoding. We also discuss an FPGA implementation of decoders for binary LDPC codes. We then explain how to combine multilevel modulation and channel coding optimally by using coded modulation. Also, we describe an LDPC-coded turbo-equalizer as a candidate for dealing simultaneously with fiber nonlinearities, PMD, and residual chromatic dispersion.

An orbital angular momentum (OAM) based LDPC-coded modulation scheme suitable for use in FSO communication is proposed. We demonstrate that the proposed scheme can operate under strong atmospheric turbulence regime and enable 100 Gb/s optical transmission while employing 10 Gb/s components. Both binary and nonbinary LDPC-coded OAM modulations are studied. In addition to providing better BER performance, the nonbinary LDPC-coded modulation reduces overall decoder complexity and latency. The nonbinary LDPC-coded OAM modulation provides a net coding gain of 9.3 dB at the BER of 10(-8). The maximum-ratio combining scheme outperforms the corresponding equal-gain combining scheme by almost 2.5 dB.

In order to achieve high-speed transmission over optical transport networks (OTNs) and maximize its throughput, we propose using a rateadaptive polarization-multiplexed coded multilevel modulation with coherent detection based on component non-binary quasi-cyclic (QC) LDPC codes. Compared to prior-art bit-interleaved LDPC-coded modulation (BI-LDPC-CM) scheme, the proposed non-binary LDPC-coded modulation (NB-LDPC-CM) scheme not only reduces latency due to symbol-instead of bit-level processing but also provides either impressive reduction in computational complexity or striking improvements in coding gain depending on the constellation size. As the paper presents, compared to its prior-art binary counterpart, the proposed NB-LDPC-CM scheme addresses the needs of future OTNs, which are achieving the target BER performance and providing maximum possible throughput both over the entire lifetime of the OTN, better.

The multidimensional channel capacity studies indicate that the employment of multiple photon degrees of freedom-such as subcarrier, amplitude, phase, polarization, and space-can improve the spectral efficiency by several orders of magnitude higher than that claimed in any fiber-optic experiment reported to date. This dramatic increase in spectral efficiency through multiple photon degrees of freedom can provide revolutionary capabilities for future optical networks. Moreover, photons can carry both spin angular momentum (SAM) associated with polarization, and orbital angular momentum (OAM) associated with the azimuthal phase of the complex electric field. Because OAM eigenstates are orthogonal, an arbitrary number of bits per photon can be transmitted in principle. The ability to generate the OAM modes, such as Bessel modes, in multimode fibers (MMFs) will allow realization of fiber-optic communication networks with ultra-high bits-per-photon efficiencies. To this end, we propose here a spatial-domain-based multidimensional coded-modulation scheme as an enabling technology for multi-Tb/s serial optical transport. To demonstrate the capabilities of the proposed scheme, we show that an eight-dimensional (8D) spatial-domain-based coded modulation scheme outperforms a prior-art 128-point 4D scheme by 3.88 dB at BER of 10(-8) while providing 120 Gb/s higher aggregate information bit rate. The proposed 8D scheme also outperforms its conventional polarization-multiplexed QAM counterpart by even a larger, and indeed striking, margin of 8.39 dB (also at the BER of 10(-8)).

We propose a scheme that can attain the same transmission bit rate as the corresponding conventional polarization-division-multiplexed (PDM) quadrature amplitude modulation (QAM) scheme while occupying lower bandwidth and, hence, achieving a higher spectral efficiency. In contrast to the conventional approach, which increases the symbol rate and thus the occupied bandwidth to transmit the redundant symbols due to forward error correction (FEC), the proposed approach expands the underlying signal constellation in size and reduces the FEC code rate accordingly to form a mechanism that can achieve coded transmission without bandwidth expansion. Such a scheme can find applications in scenarios where there exist stringent bandwidth restrictions and bandwidth expansion is not considered as a viable option. Although the idea of constellation expansion in lieu of bandwidth expansion is mainly associated with Ungerboeck's trellis-coded modulation (TCM), our proposed nonbinary low-density parity-check (LDPC)-coded modulation scheme shows that block-coded modulation schemes can also be used with expanded constellations to achieve transmission without bandwidth expansion and without resorting to TCM. Our results reveal that for small to medium constellation sizes, the proposed scheme can preserve bandwidth while not experiencing significant increase in required optical signal-tonoise ratio (OSNR). For large constellation sizes, however, to keep the increase in required OSNR at manageable levels, we propose using controlled bandwidth expansion where constellation expansion and bandwidth expansion are used simultaneously to obtain a balance between the two critical system parameters of bandwidth and required OSNR.

In this letter, we propose a four-dimensional (4-D) nonbinary low-density parity-check-coded modulation (NB-LDPC-CM) scheme suitable for beyond 100-Gb/s optical fiber communication. Incorporating spectrally efficient modulation formats to achieve high aggregate bit rates and nonbinary LDPC codes for forward error correction (FEC), the proposed scheme offers a superior advanced FEC solution for optical fiber communication systems than the prior-art bit-interleaved LDPC-coded modulation (BI-LDPC-CM) scheme. Compared to the previously reported bit error rate (BER) performance results of BI-LDPC-CM, the proposed scheme offers additional net coding gains (NCGs) of 0.29 dB, 1.17 dB, and 2.17 dB at the BER of 10 when 16-, 32-, and 64-point 4-D constellations are used, respectively.

A rate-adaptive nonbinary low-density paritycheck-coded modulation scheme is proposed to enable fiber-optic transmission at variable bit rates by adjusting the spectral efficiency (SE) of transmission while keeping the occupied bandwidth fixed. To achieve a given SE within a fixed bandwidth, both the underlying signal constellation size and the forward error correction code rate are adjusted. Using the scarce resource of bandwidth more effectively, the proposed scheme is expected to find applications in both current fixed-grid networks and in future networks that might adopt flexible grid approach.Index Terms-Forward error correction (FEC), nonbinary low-density parity-check (LDPC) codes, optical fiber communication, rate-adaptive coded modulation.

Abstract-The parity-check matrix of a nonbinary (NB) low-density parity-check (LDPC) code over Galois field GF( ) is constructed by assigning nonzero elements from GF( ) to the 1s in corresponding binary LDPC code. In this paper, we state and prove a theorem that establishes a necessary and sufficient condition that an NB matrix over GF( ) constructed by assigning nonzero elements from GF( ) to the 1s in the parity-check matrix of a binary quasi-cyclic (QC) LDPC code, must satisfy in order for its null-space to define a nonbinary QC-LDPC (NB-QC-LDPC) code. We also provide a general scheme for constructing NB-QC-LDPC codes along with some other code construction schemes targeting different goals, e.g., a scheme that can be used to construct codes for which the fast-Fourier-transform-based decoding algorithm does not contain any intermediary permutation blocks between bit node processing and check node processing steps. Via Monte Carlo simulations, we demonstrate that NB-QC-LDPC codes can achieve a net effective coding gain of 10.8 dB at an output bit error rate of 10 12 . Due to their structural properties that can be exploited during encoding/decoding and impressive error rate performance, NB-QC-LDPC codes are strong candidates for application in optical communications.