Achievable Information Rates for Coded Modulation with Hard Decision Decoding for Coherent Fiber-Optic Systems

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 al.} analyzed the AIRs for bit-wise and symbol-wise decoders considering what the authors called \emph{hard decision decoder} which, however, exploits \emph{soft information} of the transition probabilities of discrete-input discrete-output channel resulting from the hard detection. As such, the complexity of the decoder is essentially the same as the complexity of a soft decision decoder. In this paper, we analyze instead the AIRs for the standard hard decision decoder, commonly used in practice, where the decoding is based on the Hamming distance metric. We show that if standard hard decision decoding is used, bit-wise decoders yield significantly higher AIRs than symbol-wise decoders. As a result, contrary to the conclusion by Liga \emph{et al.}, binary decoders together with the BICM paradigm are preferable for spectrally-efficient fiber-optic systems. We also design binary and nonbinary staircase codes and show that, in agreement with the AIRs, binary codes yield better performance.Comment: Published in IEEE/OSA Journal of Lightwave Technology, 201

Analysis and Design of Spatially-Coupled Codes with Application to Fiber-Optical Communications

The theme of this thesis is the analysis and design of error-correcting codes that are suitable for high-speed fiber-optical communication systems. In particular, we consider two code classes. The codes in the first class are protograph-based low-density parity-check (LDPC) codes which are decoded using iterative soft-decision decoding. The codes in the second class are generalized LDPC codes with degree-2 variable nodes—henceforth referred to as generalized product codes (GPCs)—which are decoded using iterative bounded-distance decoding (BDD). Within each class, our focus is primarily on spatially-coupled codes. Spatially-coupled codes possess a convolutional structure and are characterized by a wave-like decoding behavior caused by a termination boundary effect. The contributions of this thesis can then be categorized into two topics, as outlined below.First, we consider the design of systems operating at high spectral efficiency. In particular, we study the optimization of the mapping of the coded bits to the modulation bits for a polarization-multiplexed system that is based on the bit-interleaved coded modulation paradigm. As an example, for the (protograph-based) AR4JA code family, the transmission reach can be extended by roughly up to 8% by using an optimized bit mapper, without significantly increasing the system complexity. For terminated spatially-coupled codes with long spatial length, the bit mapper optimization only results in marginal performance improvements, suggesting that a sequential allocation is close to optimal. On the other hand, an optimized allocation can significantly improve the performance of tail-biting spatially-coupled codes which do not possess an inherent termination boundary. In this case, the unequal error protection offered by the modulation bits of a nonbinary signal constellation can be exploited to create an artificial termination boundary that induces a wave-like decoding for tail-biting spatially-coupled codes.As a second topic, we study deterministically constructed GPCs. GPCs are particularly suited for high-speed applications such as optical communications due to the significantly reduced decoding complexity of iterative BDD compared to iterative soft-decision decoding of LDPC codes. We propose a code construction for GPCs which is sufficiently general to recover several well-known classes of GPCs as special cases, e.g., irregular product codes (PCs), block-wise braided codes, and staircase codes. Assuming transmission over the binary erasure channel, it is shown that the asymptotic performance of the resulting codes can be analyzed by means of a recursive density evolution (DE) equation. The DE analysis is then applied to study three different classes of GPCs: spatially-coupled PCs, symmetric GPCs, and GPCs based on component code mixtures

Analysis and Design of Spatially-Coupled Codes with Application to Fiber-Optical Communications

The theme of this thesis is the analysis and design of error-correcting codes that are suitable for high-speed fiber-optical communication systems. In particular, we consider two code classes. The codes in the first class are protograph-based low-density parity-check (LDPC) codes which are decoded using iterative soft-decision decoding. The codes in the second class are generalized LDPC codes with degree-2 variable nodes—henceforth referred to as generalized product codes (GPCs)—which are decoded using iterative bounded-distance decoding (BDD). Within each class, our focus is primarily on spatially-coupled codes. Spatially-coupled codes possess a convolutional structure and are characterized by a wave-like decoding behavior caused by a termination boundary effect. The contributions of this thesis can then be categorized into two topics, as outlined below.First, we consider the design of systems operating at high spectral efficiency. In particular, we study the optimization of the mapping of the coded bits to the modulation bits for a polarization-multiplexed system that is based on the bit-interleaved coded modulation paradigm. As an example, for the (protograph-based) AR4JA code family, the transmission reach can be extended by roughly up to 8% by using an optimized bit mapper, without significantly increasing the system complexity. For terminated spatially-coupled codes with long spatial length, the bit mapper optimization only results in marginal performance improvements, suggesting that a sequential allocation is close to optimal. On the other hand, an optimized allocation can significantly improve the performance of tail-biting spatially-coupled codes which do not possess an inherent termination boundary. In this case, the unequal error protection offered by the modulation bits of a nonbinary signal constellation can be exploited to create an artificial termination boundary that induces a wave-like decoding for tail-biting spatially-coupled codes.As a second topic, we study deterministically constructed GPCs. GPCs are particularly suited for high-speed applications such as optical communications due to the significantly reduced decoding complexity of iterative BDD compared to iterative soft-decision decoding of LDPC codes. We propose a code construction for GPCs which is sufficiently general to recover several well-known classes of GPCs as special cases, e.g., irregular product codes (PCs), block-wise braided codes, and staircase codes. Assuming transmission over the binary erasure channel, it is shown that the asymptotic performance of the resulting codes can be analyzed by means of a recursive density evolution (DE) equation. The DE analysis is then applied to study three different classes of GPCs: spatially-coupled PCs, symmetric GPCs, and GPCs based on component code mixtures

Ordered Reliability Direct Error Pattern Testing Decoding Algorithm

We introduce a novel universal soft-decision decoding algorithm for binary block codes called ordered reliability direct error pattern testing (ORDEPT). Our results, obtained for a variety of popular short high-rate codes, demonstrate that ORDEPT outperforms state-of-the-art decoding algorithms of comparable complexity such as ordered reliability bits guessing random additive noise decoding (ORBGRAND) in terms of the decoding error probability and latency. The improvements carry on to the iterative decoding of product codes and convolutional product-like codes, where we present a new adaptive decoding algorithm and demonstrate the ability of ORDEPT to efficiently find multiple candidate codewords to produce soft output

On generalized LDPC codes for ultra reliable communication

Ultra reliable low latency communication (URLLC) is an important feature in future mobile communication systems, as they will require high data rates, large system capacity and massive device connectivity [11]. To meet such stringent requirements, many error-correction codes (ECC)s are being investigated; turbo codes, low density parity check (LDPC) codes, polar codes and convolutional codes [70, 92, 38], among many others. In this work, we present generalized low density parity check (GLDPC) codes as a promising candidate for URLLC. Our proposal is based on a novel class of GLDPC code ensembles, for which new analysis tools are proposed. We analyze the trade-o_ between coding rate and asymptotic performance of a class of GLDPC codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. To incorporate both bounded distance (BD) and maximum likelihood (ML) decoding at GC nodes into our analysis without resorting to multi-edge type of degree distribution (DD)s, we propose the probabilistic peeling decoding (P-PD) algorithm, which models the decoding step at every GC node as an instance of a Bernoulli random variable with a successful decoding probability that depends on both the GC block code as well as its decoding algorithm. The P-PD asymptotic performance over the BEC can be efficiently predicted using standard techniques for LDPC codes such as Density evolution (DE) or the differential equation method. We demonstrate that the simulated P-PD performance accurately predicts the actual performance of the GLPDC code under ML decoding at GC nodes. We illustrate our analysis for GLDPC code ensembles with regular and irregular DDs. This design methodology is applied to construct practical codes for URLLC. To this end, we incorporate to our analysis the use of quasi-cyclic (QC) structures, to mitigate the code error floor and facilitate the code very large scale integration (VLSI) implementation. Furthermore, for the additive white Gaussian noise (AWGN) channel, we analyze the complexity and performance of the message passing decoder with various update rules (including standard full-precision sum product and min-sum algorithms) and quantization schemes. The block error rate (BLER) performance of the proposed GLDPC codes, combined with a complementary outer code, is shown to outperform a variety of state-of-the-art codes, for URLLC, including LDPC codes, polar codes, turbo codes and convolutional codes, at similar complexity rates.Programa Oficial de Doctorado en Multimedia y ComunicacionesPresidente: Juan José Murillo Fuentes.- Secretario: Matilde Pilar Sánchez Fernández.- Vocal: Javier Valls Coquilla