Myths and Realities of Rateless Coding

Fixed-rate and rateless channel codes are generally treated separately in the related research literature and so, a novice in the field inevitably gets the impression that these channel codes are unrelated. By contrast, in this treatise, we endeavor to further develop a link between the traditional fixed-rate codes and the recently developed rateless codes by delving into their underlying attributes. This joint treatment is beneficial for two principal reasons. First, it facilitates the task of researchers and practitioners, who might be familiar with fixed-rate codes and would like to jump-start their understanding of the recently developed concepts in the rateless reality. Second, it provides grounds for extending the use of the well-understood code design tools — originally contrived for fixed-rate codes — to the realm of rateless codes. Indeed, these versatile tools proved to be vital in the design of diverse fixed-rate-coded communications systems, and thus our hope is that they will further elucidate the associated performance ramifications of the rateless coded schemes

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

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

Reconfigurable rateless codes

We propose novel reconfigurable rateless codes, that are capable of not only varying the block length but also adaptively modify their encoding strategy by incrementally adjusting their degree distribution according to the prevalent channel conditions without the availability of the channel state information at the transmitter. In particular, we characterize a reconfigurable ratelesscode designed for the transmission of 9,500 information bits that achieves a performance, which is approximately 1 dB away from the discrete-input continuous-output memoryless channel’s (DCMC) capacity over a diverse range of channel signal-to-noise (SNR) ratios

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

Sparse graph-based coding schemes for continuous phase modulations

The use of the continuous phase modulation (CPM) is interesting when the channel represents a strong non-linearity and in the case of limited spectral support; particularly for the uplink, where the satellite holds an amplifier per carrier, and for downlinks where the terminal equipment works very close to the saturation region. Numerous studies have been conducted on this issue but the proposed solutions use iterative CPM demodulation/decoding concatenated with convolutional or block error correcting codes. The use of LDPC codes has not yet been introduced. Particularly, no works, to our knowledge, have been done on the optimization of sparse graph-based codes adapted for the context described here. In this study, we propose to perform the asymptotic analysis and the design of turbo-CPM systems based on the optimization of sparse graph-based codes. Moreover, an analysis on the corresponding receiver will be done