Multiplicatively Repeated Non-Binary LDPC Codes

We propose non-binary LDPC codes concatenated with multiplicative repetition codes. By multiplicatively repeating the (2,3)-regular non-binary LDPC mother code of rate 1/3, we construct rate-compatible codes of lower rates 1/6, 1/9, 1/12,... Surprisingly, such simple low-rate non-binary LDPC codes outperform the best low-rate binary LDPC codes so far. Moreover, we propose the decoding algorithm for the proposed codes, which can be decoded with almost the same computational complexity as that of the mother code.Comment: To appear in IEEE Transactions on Information Theor

Power and Bandwidth Efficient Coded Modulation for Linear Gaussian Channels

A scheme for power- and bandwidth-efficient communication on the linear Gaussian channel is proposed. A scenario is assumed in which the channel is stationary in time and the channel characteristics are known at the transmitter. Using interleaving, the linear Gaussian channel with its intersymbol interference is decomposed into a set of memoryless subchannels. Each subchannel is further decomposed into parallel binary memoryless channels, to enable the use of binary codes. Code bits from these parallel binary channels are mapped to higher-order near-Gaussian distributed constellation symbols. At the receiver, the code bits are detected and decoded in a multistage fashion. The scheme is demonstrated on a simple instance of the linear Gaussian channel. Simulations show that the scheme achieves reliable communication at 1.2 dB away from the Shannon capacity using a moderate number of subchannels

Design and Analysis of Nonbinary LDPC Codes for Arbitrary Discrete-Memoryless Channels

We present an analysis, under iterative decoding, of coset LDPC codes over GF(q), designed for use over arbitrary discrete-memoryless channels (particularly nonbinary and asymmetric channels). We use a random-coset analysis to produce an effect that is similar to output-symmetry with binary channels. We show that the random selection of the nonzero elements of the GF(q) parity-check matrix induces a permutation-invariance property on the densities of the decoder messages, which simplifies their analysis and approximation. We generalize several properties, including symmetry and stability from the analysis of binary LDPC codes. We show that under a Gaussian approximation, the entire q-1 dimensional distribution of the vector messages is described by a single scalar parameter (like the distributions of binary LDPC messages). We apply this property to develop EXIT charts for our codes. We use appropriately designed signal constellations to obtain substantial shaping gains. Simulation results indicate that our codes outperform multilevel codes at short block lengths. We also present simulation results for the AWGN channel, including results within 0.56 dB of the unconstrained Shannon limit (i.e. not restricted to any signal constellation) at a spectral efficiency of 6 bits/s/Hz.Comment: To appear, IEEE Transactions on Information Theory, (submitted October 2004, revised and accepted for publication, November 2005). The material in this paper was presented in part at the 41st Allerton Conference on Communications, Control and Computing, October 2003 and at the 2005 IEEE International Symposium on Information Theor

Nonbinary Spatially-Coupled LDPC Codes on the Binary Erasure Channel

We analyze the asymptotic performance of nonbinary spatially-coupled low-density parity-check (SC-LDPC) codes built on the general linear group, when the transmission takes place over the binary erasure channel. We propose an efficient method to derive an upper bound to the maximum a posteriori probability (MAP) threshold for nonbinary LDPC codes, and observe that the MAP performance of regular LDPC codes improves with the alphabet size. We then consider nonbinary SC-LDPC codes. We show that the same threshold saturation effect experienced by binary SC-LDPC codes occurs for the nonbinary codes, hence we conjecture that the BP threshold for large termination length approaches the MAP threshold of the underlying regular ensemble.Comment: Submitted to IEEE International Conference on Communications 201

Performance Prediction of Nonbinary Forward Error Correction in Optical Transmission Experiments

In this paper, we compare different metrics to predict the error rate of optical systems based on nonbinary forward error correction (FEC). It is shown that the correct metric to predict the performance of coded modulation based on nonbinary FEC is the mutual information. The accuracy of the prediction is verified in a detailed example with multiple constellation formats, FEC overheads in both simulations and optical transmission experiments over a recirculating loop. It is shown that the employed FEC codes must be universal if performance prediction based on thresholds is used. A tutorial introduction into the computation of the threshold from optical transmission measurements is also given.Comment: submitted to IEEE/OSA Journal of Lightwave Technolog

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

Applications of ordered weights in information transmission

This dissertation is devoted to a study of a class of linear codes related to a particular metric space that generalizes the Hamming space in that the metric function is defined by a partial order on the set of coordinates of the vector. We begin with developing combinatorial and linear-algebraic aspects of linear ordered codes. In particular, we define multivariate rank enumerators for linear codes and show that they form a natural set of invariants in the study of the duality of linear codes. The rank enumerators are further shown to be connected to the shape distributions of linear codes, and enable us to give a simple proof of a MacWilliams-like theorem for the ordered case. We also pursue the connection between linear codes and matroids in the ordered case and show that the rank enumerator can be thought of as an instance of the classical matroid invariant called the Tutte polynomial. Finally, we consider the distributions of support weights of ordered codes and their expression via the rank enumerator. Altogether, these results generalize a group of well-known results for codes in the Hamming space to the ordered case. Extending the research in the first part, we define simple probabilistic channel models that are in a certain sense matched to the ordered distance, and prove several results related to performance of linear codes on such channels. In particular, we define ordered wire-tap channels and establish several results related to the use of linear codes for reliable and secure transmission in such channel models. In the third part of this dissertation we study polar coding schemes for channels with nonbinary input alphabets. We construct a family of linear codes that achieve the capacity of a nonbinary symmetric discrete memoryless channel with input alphabet of size q=2^r, r=2,3,.... A new feature of the coding scheme that arises in the nonbinary case is related to the emergence of several extremal configurations for the polarized data symbols. We establish monotonicity properties of the configurations and use them to show that total transmission rate approaches the symmetric capacity of the channel. We develop these results to include the case of ``controlled polarization'' under which the data symbols polarize to any predefined set of extremal configurations. We also outline an application of this construction to data encoding in video sequences of the MPEG-2 and H.264/MPEG-4 standards

Rate-Adaptive Coded Modulation for Fiber-Optic Communications

Rate-adaptive optical transceivers can play an important role in exploiting the available resources in dynamic optical networks, in which different links yield different signal qualities. We study rate-adaptive joint coding and modulation, often called coded modulation (CM), addressing non-dispersion-managed (non-DM) links, exploiting recent advances in channel modeling of these links. We introduce a four-dimensional CM scheme, which shows a better tradeoff between digital signal processing complexity and transparent reach than existing methods. We construct a rate-adaptive CM scheme combining a single low-density parity-check code with a family of three signal constellations and using probabilistic signal shaping. We evaluate the performance of the proposed CM scheme for single-channel transmission through long-haul non-DM fiber-optic systems with electronic chromatic-dispersion compensation. The numerical results demonstrate improvement of spectral efficiency over a wide range of transparent reaches, an improvement over 1 dB compared to existing methods