On the Optimality of Single-label and Multi-label Neural Network Decoders

We investigate the design of two neural network (NN) architectures recently proposed as decoders for forward error correction: the so-called single-label NN (SLNN) and multi-label NN (MLNN) decoders. These decoders have been reported to achieve near-optimal codeword- and bit-wise performance, respectively. Results in the literature show near-optimality for a variety of short codes. In this paper, we analytically prove that certain SLNN and MLNN architectures can, in fact, always realize optimal decoding, regardless of the code. These optimal architectures and their binary weights are shown to be defined by the codebook, i.e., no training or network optimization is required. Our proposed architectures are in fact not NNs, but a different way of implementing the maximum likelihood decoding rule. Optimal performance is numerically demonstrated for Hamming $(7,4)$, Polar $(16,8)$, and BCH $(31,21)$ codes. The results show that our optimal architectures are less complex than the SLNN and MLNN architectures proposed in the literature, which in fact only achieve near-optimal performance. Extension to longer codes is still hindered by the curse of dimensionality. Therefore, even though SLNN and MLNN can perform maximum likelihood decoding, such architectures cannot be used for medium and long codes.

On the Design and Performance of Machine Learning Based Error Correcting Decoders

This paper analyzes the design and competitiveness of four neural network (NN) architectures recently proposed as decoders for forward error correction (FEC) codes. We first consider the so-called single-label neural network (SLNN) and the multi-label neural network (MLNN) decoders which have been reported to achieve near maximum likelihood (ML) performance. Here, we show analytically that SLNN and MLNN decoders can always achieve ML performance, regardless of the code dimensions -- although at the cost of computational complexity -- and no training is in fact required. We then turn our attention to two transformer-based decoders: the error correction code transformer (ECCT) and the cross-attention message passing transformer (CrossMPT). We compare their performance against traditional decoders, and show that ordered statistics decoding outperforms these transformer-based decoders. The results in this paper cast serious doubts on the application of NN-based FEC decoders in the short and medium block length regime.

Band-ESS: Streaming Enumerative Coding with Applications to Probabilistic Shaping

Probabilistic amplitude shaping (PAS) is on track to become the de facto coded modulation standard for communication systems aiming to operate close to channel capacity at high transmission rates. The essential component of PAS that breeds this widespread interest is the amplitude shaping block, through which the channel input distribution is controlled. This block is responsible for converting bit strings into amplitude sequences with certain properties, e.g., fixed composition, limited energy, limited energy variation, etc. Recently, band-trellis enumerative sphere shaping (B-ESS) was introduced as an amplitude shaping technique that achieves limited energy variations which is useful in optical communication scenarios. B-ESS operates based on a trellis diagram in which sequences with high energy variations are pruned. In this work, we study the implementation of B-ESS. We first show that thanks to the trellis structure obtained by this pruning, B-ESS can be implemented with very low storage complexity. The trellis computation is shown to be reduced to a set of recursive multiplications with a scalar factor. Then we show that this scalar factor can be adjusted such that the trellis computation is further simplified and realized with only binary shifts. This shift-based B-ESS (1) can be implemented for arbitrarily long blocklengths without incurring an increase in complexity, and (2) can operate in a streaming mode similar to convolutional coding.

Temporal Properties of Enumerative Shaping: Autocorrelation and Energy Dispersion Index

We study the effective SNR behavior of various enumerative amplitude shaping algorithms. We show that their relative behavior can be explained via the temporal autocorrelation function or via the energy dispersion index.

On Optimum Enumerative Sphere Shaping Blocklength at Different Symbol Rates for the Nonlinear Fiber Channel

We show that a 0.9 dB SNR improvement can be obtained via short-blocklength enumerative sphere shaping for single-span transmission at 56 GBd. This gain vanishes for higher symbol rates and a larger number of spans.

Log-CCDM: Distribution Matching via Multiplication-free Arithmetic Coding

Recent years have seen renewed attention to arithmetic coding (AC). This is thanks to the use of AC for distribution matching (DM) to control the channel input distribution in probabilistic amplitude shaping. There are two main problems inherent to AC: (1) its required arithmetic precision grows linearly with the input length, and (2) high-precision multiplications and divisions are required. Here, we introduce a multiplication-free AC-based DM technique via three lookup tables (LUTs) which solves both problems above. These LUTs are used to approximate the high-precision multiplications and divisions by additions and subtractions. The required precision of our approach is shown to grow logarithmically with the input length. We prove that this approximate technique maintains the invertibility of DM. At an input length of 1024 symbols, the proposed technique achieves negligible rate loss ($<0.01$ bit/sym) against the full-precision DM, while requiring less than 4 kilobytes of storage.

Introducing 4D Geometric Shell Shaping

Four dimensional geometric shell shaping (4D-GSS) is introduced and evaluated for reach increase and nonlinearity tolerance in terms of achievable information rates and post-FEC bit-error rate. A format is designed with a spectral efficiency of 8 bit/4D-sym and is compared against polarization-multiplexed 16QAM (PM-16QAM) and probabilistically shaped PM-16QAM (PS-PM-16QAM) in a 400ZR-compatible transmission setup with high amount of nonlinearities. Numerical simulations for a single-span, single-channel show that 4D-GSS achieves increased nonlinear tolerance and reach increase against PM-16QAM and PS-PM-16QAM when optimized for bit-metric decoding (RBMD). In terms of RBMD, gains are small with a reach increase of 1.6% compared to PM-16QAM. When optimizing for mutual information, a larger reach increase of 3% is achieved compared to PM-16QAM. Moreover, the introduced GSS scheme provides a scalable framework for designing well-structured 4D modulation formats with low complexity.

4D Geometric Shell Shaping with Applications to 400ZR

Geometric shell shaping is introduced and evaluated for reach increase and nonlinearity tolerance in terms of MI against PM-16QAM and PS-PM-16QAM in a 400ZR compatible transmission setup.

Mitigating Nonlinear Interference by Limiting Energy Variations in Sphere Shaping

Band-trellis enumerative sphere shaping is proposed to decrease the energy variations in channel input sequences. Against sphere shaping, 0.74 dB SNR gain and up to 9% increase in data rates are demonstrated for single-span systems.

Low-Complexity Geometrical Shaping for 4D Modulation Formats via Amplitude Coding

Signal shaping is vital to approach Shannon's capacity, yet it is challenging to implement at very high speeds. For example, probabilistic shaping often requires arithmetic coding to realize the target distribution. Geometric shaping requires look-up tables to store the constellation points. In this paper, we propose a four-dimensional amplitude coding (4D-AC) geometrical shaper architecture. The proposed architecture can generate in real time geometrically shaped 4D formats via simple logic circuit operations and two conventional quadrature amplitude modulation (QAM) modulators. This paper describes the 4D-AC used in generating approximated versions of two recently proposed 4D orthant symmetric modulation formats with spectral efficiencies of 6 bit/4D-sym and 7 bit/4D-sym, respectively. Numerical results show losses below 0.05 dB when compared against the baseline formats.