On The Design Of Physical Layer Rateless Codes

Codes that are capable of generating any number of encoded symbols from a given number of source symbols are called rateless codes. Luby transform (LT) codes are the first practical realization of rateless codes while Raptor codes are constructed by serially concatenating LT codes with high-rate outer low-density parity-check (LDPC) codes. Although these codes were originally developed for binary erasure channel (BEC), due to their rateless feature, they are being investigated and designed for their use in noisy channels. It is known that LT codes are the irregular non-systematic rateless counterpart of low-density generator-matrix (LDGM) codes. Therefore, the first part of our work is focused on LDGM codes and their serially concatenated scheme called serially concatenated LDGM (SCLDGM) codes. Though single LDGM codes are asymptotically bad codes, the SCLDGM codes are known to perform close to the Shannon limit. We first study the asymptotic behaviour of LDGM codes using a discretized density evolution method. We then show that the DDE method can be used in two-steps to provide the detailed asymptotic performance analysis of SCLDGM codes. We also provide the detailed error-floor analysis of both the LDGM and SCLDGM codes. We also prove a necessary condition for the successful decoding of such concatenated codes under sum-product (SP) decoding in binary input additive white Gaussian noise (BIAWGN) channels. Based on this necessary condition, we then develop a DDE-based optimization approach which can be used to optimize such concatenated codes in general. We present both the asymptotic performance and simulation results of our optimized SCLDGM codes that perform within 0.26 dB to the Shannon limit in BIAWGN channels. Secondly, we focus on the asymptotic analysis and optimization design of LT and Raptor codes over BIAWGN channels. We provide the exact asymptotic performance of LT codes using the DDE method. We apply the concept of the two-step DDE method to the Raptor codes and obtain their exact asymptotic performance in BIAWGN channels. We show that the existing Raptor codes using solely the same output degree distribution can perform within 0.4 dB to the Shannon limit for various realized code-rates. We then develop a DDE-based optimization technique to optimally design such physical layer Raptor codes. Our optimized Raptor codes are shown to perform within 0.2 dB to the Shannon limit for most of the realized code-rates. We also provide the asymptotic curves, decoding thresholds, and simulation results showing that our optimized Raptor codes outperform the existing Raptor codes in BIAWGN channels. Finally, we present the asymptotic analysis and optimization design of systematic version of these codes namely systematic LT and systematic Raptor codes as well

Rewriting Flash Memories by Message Passing

This paper constructs WOM codes that combine rewriting and error correction for mitigating the reliability and the endurance problems in flash memory. We consider a rewriting model that is of practical interest to flash applications where only the second write uses WOM codes. Our WOM code construction is based on binary erasure quantization with LDGM codes, where the rewriting uses message passing and has potential to share the efficient hardware implementations with LDPC codes in practice. We show that the coding scheme achieves the capacity of the rewriting model. Extensive simulations show that the rewriting performance of our scheme compares favorably with that of polar WOM code in the rate region where high rewriting success probability is desired. We further augment our coding schemes with error correction capability. By drawing a connection to the conjugate code pairs studied in the context of quantum error correction, we develop a general framework for constructing error-correction WOM codes. Under this framework, we give an explicit construction of WOM codes whose codewords are contained in BCH codes.Comment: Submitted to ISIT 201

Low Density Graph Codes And Novel Optimization Strategies For Information Transfer Over Impaired Medium

Effective methods for information transfer over an imperfect medium are of great interest. This thesis addresses the following four topics involving low density graph codes and novel optimization strategies.Firstly, we study the performance of a promising coding technique: low density generator matrix (LDGM) codes. LDGM codes provide satisfying performance while maintaining low encoding and decoding complexities. In the thesis, the performance of LDGM codes is extracted for both majority-rule-based and sum-product iterative decoding algorithms. The ultimate performance of the coding scheme is revealed through distance spectrum analysis. We derive the distance spectral for both LDGM codes and concatenated LDGM codes. The results show that serial-concatenated LDGM codes deliver extremely low error-floors. This work provides valued information for selecting the parameters of LDGM codes. Secondly, we investigate network-coding on relay-assisted wireless multiple access (WMA) networks. Network-coding is an effective way to increase robustness and traffic capacity of networks. Following the framework of network-coding, we introduce new network codes for the WMA networks. The codes are constructed based on sparse graphs, and can explore the diversities available from both the time and space domains. The data integrity from relays could be compromised when the relays are deployed in open areas. For this, we propose a simple but robust security mechanism to verify the data integrity.Thirdly, we study the problem of bandwidth allocation for the transmission of multiple sources of data over a single communication medium. We aim to maximize the overall user satisfaction, and formulate an optimization problem. Using either the logarithmic or exponential form of satisfaction function, we derive closed-form optimal solutions, and show that the optimal bandwidth allocation for each type of data is piecewise linear with respect to the total available bandwidth. Fourthly, we consider the optimization strategy on recovery of target spectrum for filter-array-based spectrometers. We model the spectrophotometric system as a communication system, in which the information content of the target spectrum is passed through distortive filters. By exploiting non-negative nature of spectral content, a non-negative least-square optimal criterion is found particularly effective. The concept is verified in a hardware implemen

Polar Coding Schemes for Cooperative Transmission Systems

: In this thesis, a serially-concatenated coding scheme with a polar code as the outer code and a low density generator matrix (LDGM) code as the inner code is firstly proposed. It is shown that that the proposed scheme provides a method to improve significantly the low convergence of polar codes and the high error floor of LDGM codes while keeping the advantages of both such as the low encoding and decoding complexity. The bit error rate results show that the proposed scheme by reasonable design have the potential to approach a performance close to the capacity limit and avoid error floor effectively. Secondly, a novel transmission protocol based on polar coding is proposed for the degraded half-duplex relay channel. In the proposed protocol, the relay only needs to forward a part of the decoded source message that the destination needs according to the exquisite nested structure of polar codes. It is proved that the scheme can achieve the capacity of the half-duplex relay channel while enjoying low encoding/decoding complexity. By modeling the practical system, we verify that the proposed scheme outperforms the conventional scheme designed by low-density parity-check codes by simulations. Finally, a generalized partial information relaying protocol is proposed for degraded multiple-relay networks with orthogonal receiver components (MRN-ORCs). In such a protocol, each relay node decodes the received source message with the help of partial information from previous nodes and re-encodes part of the decoded message for transmission to satisfy the decoding requirements for the following relay node or the destination node. For the design of polar codes, the nested structures are constructed based on this protocol and the information sets corresponding to the partial messages forwarded are also calculated. It is proved that the proposed scheme achieves the theoretical capacity of the degraded MRN-ORCs while still retains the low-complexity feature of polar codes

The Road From Classical to Quantum Codes: A Hashing Bound Approaching Design Procedure

Powerful Quantum Error Correction Codes (QECCs) are required for stabilizing and protecting fragile qubits against the undesirable effects of quantum decoherence. Similar to classical codes, hashing bound approaching QECCs may be designed by exploiting a concatenated code structure, which invokes iterative decoding. Therefore, in this paper we provide an extensive step-by-step tutorial for designing EXtrinsic Information Transfer (EXIT) chart aided concatenated quantum codes based on the underlying quantum-to-classical isomorphism. These design lessons are then exemplified in the context of our proposed Quantum Irregular Convolutional Code (QIRCC), which constitutes the outer component of a concatenated quantum code. The proposed QIRCC can be dynamically adapted to match any given inner code using EXIT charts, hence achieving a performance close to the hashing bound. It is demonstrated that our QIRCC-based optimized design is capable of operating within 0.4 dB of the noise limit

Spatially-Coupled Precoded Rateless Codes

Raptor codes are rateless codes that achieve the capacity on the binary erasure channels. However the maximum degree of optimal output degree distribution is unbounded. This leads to a computational complexity problem both at encoders and decoders. Aref and Urbanke investigated the potential advantage of universal achieving-capacity property of proposed spatially-coupled (SC) low-density generator matrix (LDGM) codes. However the decoding error probability of SC-LDGM codes is bounded away from 0. In this paper, we investigate SC-LDGM codes concatenated with SC low-density parity-check codes. The proposed codes can be regarded as SC Hsu-Anastasopoulos rateless codes. We derive a lower bound of the asymptotic overhead from stability analysis for successful decoding by density evolution. The numerical calculation reveals that the lower bound is tight. We observe that with a sufficiently large number of information bits, the asymptotic overhead and the decoding error rate approach 0 with bounded maximum degree

Erasure Based Soft Decision Decoding for Optical Transport Networks

A concatenated soft decision forward error correcting (FEC) coding scheme with an inner low density generator matrix (LDGM) code and an outer product code (PC) is considered for applications in high-speed optical communications. First, we evaluate the performance of various choices of inner and outer codes when the inner decoder uses a soft-decision decoder and the outer decoder is a hard-decision error-only decoder. Then, we evaluate the performance of the concatenated coding scheme when erasures are introduced at the output of the inner decoder and the outer decoder is an errors-and-erasures decoder. An exact expression for the number of errors and erasures that are guaranteed to be decoded by an iterative decoder is derived. Then, an approach for deriving an approximation of the error probability for the erasures decoder is proposed. Using this approach, optimal thresholds can be chosen for declaring erasures at the output of the inner decoder. It is shown that the codeword error rate for the error and erasures iterative decoder can be better than that of errors only iterative decoder in the error floor region. It is also shown that simulation results are in close agreement with the mathematical approximations developed at error floors

Design of serially-concatenated LDGM codes

[Resumen] Since Shannon demonstrated in 1948 the feasibility of achieving an arbitrarily low error probability in a communications system provided that the transmission rate was kept below a certain limit, one of the greatest challenges in the realm of digital communications and, more specifically, in the channel coding field, has been finding codes that are able to approach this limit as much as possible with a reasonable encoding and decoding complexity, However, it was not until 1993, when Berrou et al. presented the turbo codes, that a coding scheme capable of performing at less than 1dB from Shannon's limit with an extremely low error probability was found. The idea on which these codes are based is the iterative decoding of concatenated components that exchange information about the transmitted bits, which is known as the "turbo principle". The generalization of this idea led in 1995 to the rediscovery of LDPC (Low Density Parity Check) codes, proposed for the first time by Gallager in the 60s. LDPC codes are linear block codes with a sparse parity check matrix that are able to surpass the performance of turbo codes with a smaller decoding complexity. However, due to the fact that the generator matrix of general LDPC codes is not sparse, their encoding complexity can be excessively high. LDGM (Low Density Generator Matrix) codes, a particular case of LDPC codes, are codes with a sparse generator matrix, thanks to which they present a lower encoding complexity. However, except for the case of very high rate codes, LDGM codes are "bad", i.e., they have a non-zero error probability that is independent of the code block length. More recently, IRA (Irregular Repeat-Accumulated) codes, consisting of the serial concatenation of a LDGM code and an accumulator, have been proposed. IRA codes are able to get close to the performance of LDPC codes with an encoding complexity similar to that of LDGM codes. In this thesis we explore an alternative to IRA codes consisting in the serial concatenation of two LDGM codes, a scheme that we will denote SCLDGM (Serially-Concatenated Low-Density Generator Matrix). The basic premise of SCLDGM codes is that an inner code of rate close to the desired transmission rate fixes most of the errors, and an external code of rate close to one corrects the few errors that result from decoding the inner code. For any of these schemes to perform as close as possible to the capacity limit it is necessary to determine the code parameters that best fit the channel over which the transmission will be done. The two techniques most commonly used in the literature to optimize LDPC codes are Density Evolution (DE) and EXtrinsic Information Transfer (EXIT) charts, which have been employed to obtain optimized codes that perform at a few tenths of a decibel of the AWGN channel capacity. However, no optimization techniques have been presented for SCLDGM codes, which so far have been designed heuristically and therefore their performance is far from the performance achieved by IRA and LDPC codes. Other of the most important advances that have occurred in recent years is the utilization of multiple antennas at the trasmitter and the receiver, which is known as MIMO (Multiple-Input Multiple-Output) systems. Telatar showed that the channel capacity in these kind of systems scales linearly with the minimum number of transmit and receive antennas, which enables us to achieve spectral efficiencies far greater than with systems with a single transmit and receive antenna (or Single Input Single Output (SISO) systems). This important advantage has attracted a lot of attention from the research community, and has caused that many of the new standards, such as WiMax 802.16e or WiFi 802.11n, as well as future 4G systems are based on MIMO systems. The main problem of MIMO systems is the high complexity of optimum detection, which grows exponentially with the number of transmit antennas and the number of modulation levels. Several suboptimum algorithms have been proposed to reduce this complexity, most notably the SIC-MMSE (Soft-Interference Cancellation Minimum Mean Square Error) and spherical detectors. Another major issue is the high complexity of the channel estimation, due to the large number of coefficients which determine it. There are techniques, such as Maximum-Likelihood-Expectation-Maximization (ML-EM), that have been successfully applied to estimate MIMO channels but, as in the case of detection, they suffer from the problem of a very high complexity when the number of transmit antennas or the size of the constellation increase. The main objective of this work is the study and optimization of SCLDGM codes in SISO and MIMO channels. To this end, we propose an optimization method for SCLDGM codes based on EXIT charts that allow these codes to exceed the performance of IRA codes existing in the literature and get close to the performance of LDPC codes, with the advantage over the latter of a lower encoding complexity. We also propose optimized SCLDGM codes for both spherical and SIC-MMSE suboptimal MIMO detectors, constituting a system that is capable of approaching the capacity limits of MIMO channels with a low complexity encoding, detection and decoding. We analyze the BICM (Bit-Interleaved Coded Modulation) scheme and the concatenation of SCLDGM codes with Space-Time Codes (STC) in ergodic and quasi-static MIMO channels. Furthermore, we explore the combination of these codes with different channel estimation algorithms that will take advantage of the low complexity of the suboptimum detectors to reduce the complexity of the estimation process while keeping a low distance to the capacity limit. Finally, we propose coding schemes for low rates involving the serial concatenation of several LDGM codes, reducing the complexity of recently proposed schemes based on Hadamard codes