Finite Length Analysis of Rateless Codes and Their Application in Wireless Networks

Mobile communication systems are undergoing revolutionary developments as a result of the rapidly growing demands for high data rates and reliable communication connections. The key features of the next-generation mobile communication systems are provision of high-speed and robust communication links. However, wireless communications still need to address the same challenge–unreliable communication connections, arising from a number of causes including noise, interference, and distortion because of hardware imperfections or physical limitations. Forwarding error correction (FEC) codes are used to protect source information by adding redundancy. With FEC codes, errors among the transmitted message can be corrected by the receiver. Recent work has shown that, by applying rateless codes (a class of FEC codes), wireless transmission efficiency and reliability can be dramatically improved. Unlike traditional codes, rateless codes can adapt to different channel conditions. Rateless codes have been widely used in many multimedia broadcast/multicast applications. Among the known rate- less codes, two types of codes stand out: Luby transform (LT) codes and Raptor codes. However, our understanding of LT codes and Raptor codes is still in- complete due to the lack of complete theoretical analysis on the decoding error performance of these codes. Particularly, this thesis focuses on the decoding error performance of these codes under maximum-likelihood (ML) decoding, which provides a benchmark on the optimum system performance for gauging other decoding schemes. In this thesis, we discuss the effectiveness of rateless codes in terms of the success probability of decoding. It is defined as the probability that all source symbols can be successfully decoded with a given number of success- fully received coded symbols under ML decoding. This thesis provides a detailed mathematical analysis on the rank profile of general LT codes to evaluate the decoding success probability of LT codes under ML decoding. Furthermore, by analyzing the rank of the product of two random coefficient matrices, this thesis derived bounds on the decoding success probability of Raptor codes with a systematic low-density generator matrix (LDGM) code as the pre-code under ML decoding. Additionally, by resorting to stochastic geometry analysis, we develop a rateless codes based broadcast scheme. This scheme allows a base station (BS) to broadcast a given number of symbols to a large number of users, without user acknowledgment, while being able to pro- vide a performance guarantee on the probability of successful delivery. Further, the BS has limited statistical information about the environment including the spatial distribution of users (instead of their exact locations and number) and the wireless propagation model. Based on the analysis of finite length LT codes and Raptor codes, an upper and a lower bound on the number of transmissions required to meet the performance requirement are obtained. The technique and analysis developed in this thesis are useful for designing efficient and reliable wireless broadcast strategies. It is of interest to implement rateless codes into modern communication systems

On Linear Operator Channels over Finite Fields

Motivated by linear network coding, communication channels perform linear operation over finite fields, namely linear operator channels (LOCs), are studied in this paper. For such a channel, its output vector is a linear transform of its input vector, and the transformation matrix is randomly and independently generated. The transformation matrix is assumed to remain constant for every T input vectors and to be unknown to both the transmitter and the receiver. There are NO constraints on the distribution of the transformation matrix and the field size. Specifically, the optimality of subspace coding over LOCs is investigated. A lower bound on the maximum achievable rate of subspace coding is obtained and it is shown to be tight for some cases. The maximum achievable rate of constant-dimensional subspace coding is characterized and the loss of rate incurred by using constant-dimensional subspace coding is insignificant. The maximum achievable rate of channel training is close to the lower bound on the maximum achievable rate of subspace coding. Two coding approaches based on channel training are proposed and their performances are evaluated. Our first approach makes use of rank-metric codes and its optimality depends on the existence of maximum rank distance codes. Our second approach applies linear coding and it can achieve the maximum achievable rate of channel training. Our code designs require only the knowledge of the expectation of the rank of the transformation matrix. The second scheme can also be realized ratelessly without a priori knowledge of the channel statistics.Comment: 53 pages, 3 figures, submitted to IEEE Transaction on Information Theor

Beyond the Min-Cut Bound: Deterministic Network Coding for Asynchronous Multirate Broadcast

In a single hop broadcast packet erasure network, we demonstrate that it is possible to provide multirate packet delivery outside of what is given by the network min-cut. This is achieved by using a deterministic non-block-based network coding scheme, which allows us to sidestep some of the limitations put in place by the block coding model used to determine the network capacity. Under the network coding scheme we outline, the sender is able to transmit network coded packets above the channel rate of some receivers, while ensuring that they still experience nonzero delivery rates. Interestingly, in this generalised form of asynchronous network coded broadcast, receivers are not required to obtain knowledge of all packets transmitted so far. Instead, causal feedback from the receivers about packet erasures is used by the sender to determine a network coded transmission that will allow at least one, but often multiple receivers, to deliver their next needed packet. Although the analysis of deterministic coding schemes is generally a difficult problem, by making some approximations we are able to obtain tractable estimates of the receivers' delivery rates, which are shown to match reasonably well with simulation. Using these estimates, we design a fairness algorithm that allocates the sender's resources so all receivers will experience fair delivery rate performance