Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

国際会議開催報告:2013 IEEE International Symposium on Information Theory

An investigation is made of relays whose reliability can be described in simple terms by means of probabilities. It is shown that by using a sufficiently large number of these relays in the proper manner, circuits can be built which are arbitrarily reliable, regardless of how unreliable the original relays are. Various properties of these circuits are elucidated.

Reliable Circuits Using Less Reliable Relays

Concentration inequalities have been the subject of exciting developments during the last two decades, and have been intensively studied and used as a powerful tool in various areas. These include convex geometry, functional analysis, statistical physics, mathematical statistics, pure and applied probability theory (e.g., concentration of measure phenomena in random graphs, random matrices, and percolation), information theory, theoretical computer science, learning theory, and dynamical systems.This monograph focuses on some of the key modern mathematical tools that are used for the derivation of concentration inequalities, on their links to information theory, and on their various applications to communications and coding. In addition to being a survey, this monograph also includes various new recent results derived by the authors.The first part of the monograph introduces classical concentration inequalities for martingales, aswell as some recent refinements and extensions. The power and versatility of the martingale approach is exemplified in the context of codes defined on graphs and iterative decoding algorithms, as well as codes for wireless communication.The second part of the monograph introduces the entropy method, an information-theoretic technique for deriving concentration inequalities for functions of many independent random variables. The basic ingredients of the entropy method are discussed first in conjunction with the closely related topic of logarithmic Sobolev inequalities, which are typical of the so-called functional approach to studying the concentration of measure phenomenon. The discussion on logarithmic Sobolev inequalities is complemented by a related viewpoint based on probability in metric spaces. This viewpoint centers around the so-called transportation-cost inequalities, whose roots are in information theory. Some representative results on concentration for dependent random variables are briefly summarized, with emphasis on their connections to the entropy method. Finally, we discuss several applications of the entropy method and related information-theoretic tools to problems in communications and coding. These include strong converses, empirical distributions of good channel codes with non-vanishing error probability, and an information-theoretic converse for concentration of measure.

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Concentration of Measure Inequalities in Information Theory, Communications and Coding

Introduction to finite fields and their applications: Factorization of Polynomials

A random access scheme is a fundamental scenario in which users transmit through a shared channel and cannot coordinate with each other. In recent years, successive interference cancellation (SIC) is introduced into the random access scheme. It is possible to decode transmitted packets using collided packets by the SIC. The coded slotted ALOHA (CSA) is a random access scheme using the SIC. The CSA encodes each packet using a local code prior to transmission. It is known that the CSA achieves excellent throughput. On the other hand, it is reported that shift operation improves the decoding performance for packet-oriented erasure correcting coding system. In this paper, we propose a random access scheme which applies the shift operation to the CSA. Numerical examples show that our proposed random access scheme achieves better throughput and packet loss rate than the CSA.

Shifted Coded Slotted ALOHA

In this paper, we propose a decoding algorithm which lowers decoding erasure rates in the error floor regions for non-binary low-density parity-check codes transmitted over the binary erasure channels. This decoding algorithm is a combination with belief propagation (BP) decoding and maximum a posteriori (MAP) decoding on zigzag cycles, which cause decoding erasures in the error floor region. We show that MAP decoding on the zigzag cycles is realized by means of a message passing algorithm. A simulation result shows that the decoding erasure rates in the error floor regions by the proposed decoding algorithm are lower than those by the BP decoder.

Message passing algorithm with MAP decoding on zigzag cycles for non-binary LDPC codes

In this paper, we investigate the error floors of non-binary low-density parity-check (LDPC) codes transmitted over the memoryless binary-input output-symmetric (MBIOS) channels. We provide a necessary and sufficient condition for successful decoding of zigzag cycle codes over the MBIOS channel by the belief propagation decoder. We consider an expurgated ensemble of non-binary LDPC codes by using the above necessary and sufficient condition, and hence exhibit lower error floors. Finally, we show lower bounds of the error floors for the expurgated LDPC code ensembles over the MBIOS channels.

Analysis of Error Floors of Non-binary LDPC Codes over MBIOS Channel

An Improvement of Non-Binary Single b -Burst of Insertion/Deletion Correcting Code

Fountain codes are erasure correcting codes realizing reliable communication on multicasting. In a previous work, we proposed a fountain code, named zigzag decodable fountain (ZDF) code, and showed that the ZDF code outperforms Raptor codes in term of the decoding performance. However, the conventional decoding algorithm for the ZDF code, which is regarded as a combination of packet-wise peeling algorithm and bit-wise peeling algorithm, requires a large number of iterations. The main purpose of this research is the reduction of the number of decoding iterations for the ZDF codes. The tree-structure expectation propagation (TEP) is a decoding algorithm for the low-density parity-check codes over the binary erasure channels and an extension of peeling algorithm. In this paper, we extend the TEP for the ZDF codes and propose a decoding algorithm combined the packet-wise TEP with bit-wise peeling algorithm. Simulation results show that the proposed decoding algorithm reduces the number of decoding iterations as compared with the previous decoding algorithm without loss of decoding performance.

Takayuki Nozaki

Papers

Shifted Coded Slotted ALOHA

Message passing algorithm with MAP decoding on zigzag cycles for non-binary LDPC codes

Analysis of Error Floors of Non-binary LDPC Codes over MBIOS Channel

An Improvement of Non-Binary Single b -Burst of Insertion/Deletion Correcting Code

Reduction of decoding iterations for zigzag decodable fountain codes