Fountain Codes under Maximum Likelihood Decoding

This dissertation focuses on fountain codes under maximum likelihood (ML) decoding. First LT codes are considered under a practical and widely used ML decoding algorithm known as inactivation decoding. Different analysis techniques are presented to characterize the decoding complexity. Next an upper bound to the probability of decoding failure of Raptor codes under ML decoding is provided. Then, the distance properties of an ensemble of fixed-rate Raptor codes with linear random outer codes are analyzed. Finally, a novel class of fountain codes is presented, which consists of a parallel concatenation of a block code with a linear random fountain code.Comment: PhD Thesi

Bounds on the Error Probability of Raptor Codes under Maximum Likelihood Decoding

In this paper upper and lower bounds on the probability of decoding failure under maximum likelihood decoding are derived for different (nonbinary) Raptor code constructions. In particular four different constructions are considered; (i) the standard Raptor code construction, (ii) a multi-edge type construction, (iii) a construction where the Raptor code is nonbinary but the generator matrix of the LT code has only binary entries, (iv) a combination of (ii) and (iii). The latter construction resembles the one employed by RaptorQ codes, which at the time of writing this article represents the state of the art in fountain codes. The bounds are shown to be tight, and provide an important aid for the design of Raptor codes.Comment: Submitted for revie

Inactivation Decoding of LT and Raptor Codes: Analysis and Code Design

In this paper we analyze LT and Raptor codes under inactivation decoding. A first order analysis is introduced, which provides the expected number of inactivations for an LT code, as a function of the output distribution, the number of input symbols and the decoding overhead. The analysis is then extended to the calculation of the distribution of the number of inactivations. In both cases, random inactivation is assumed. The developed analytical tools are then exploited to design LT and Raptor codes, enabling a tight control on the decoding complexity vs. failure probability trade-off. The accuracy of the approach is confirmed by numerical simulations.Comment: Accepted for publication in IEEE Transactions on Communication

The design of degree distribution for distributed fountain codes in wireless sensor networks

In this paper, we first analyse bit error rate (BER) bounds of the distributed network coding (DNC) scheme based on the Luby-transform (LT) codes, which is a class of fountain codes, for wireless sensor networks (WSNs). Then we investigate the effect from two parameters of the degree distributions, i.e., the degree value and the proportion of odd degree, to the performance of the LT-based DNC scheme. Based on the analysis and investigation results, a degree distribution design criteria is proposed for the DNC scheme based on fountain codes over Rayleigh fading channels. We compare the performance of the DNC scheme based on fountain codes using degree distributions designed in this paper with other schemes given in the literature. The comparison results show that the degree distributions designed by using the proposed criteria have better performance

Basis-Finding Algorithm for Decoding Fountain Codes for DNA-Based Data Storage

In this paper, we consider the decoding of fountain codes where the received symbols may have errors. It is motivated by the application of fountain codes in the DNA-based data storage systems where the inner code decoding, which generally has undetectable errors, is performed before the outer fountain code decoding. We propose a novel and efficient decoding algorithm, namely the basis-finding algorithm (BFA). The key idea of the BFA is to find a basis of the received symbols, and then use the most reliable basis elements to recover the source symbols with the inactivation decoding. Gaussian elimination is used to find the basis and to identify the most reliable basis elements. As a result, the BFA has polynomial time complexity. For random fountain codes, we are able to derive some theoretical bounds for the frame error rate (FER) of the BFA. Extensive simulations with Luby transform (LT) codes show that for the scenario we consider, the BFA has significantly better FER than the belief propagation (BP) algorithm except when the overhead is extremely large.Comment: 13 page

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