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 code is bounded single insertion/deletion correcting if a decoder corrects a single insertion or single deletion with side information about a range of positions occurring an inserted symbol or deleted symbol. This paper constructs two bounded single insertion/deletion correcting codes and gives decoding algorithms for these. Moreover, we evaluate the number of codewords of these codes. As a result, the constructed codes have larger cardinalities than the existing ones.

Bounded Single Insertion/Deletion Correcting Codes

In this paper, we analyze (2, k)-regular non-binary low-density parity-check codes over the binary erasure channels. We propose a method to improve the error floors by optimizing labels in zigzag cycles in the Tanner graph. We analyze the error floors for codes designed by the proposed optimization method and show that the error floors are decreasing in the size of Galois field.

Error floors of non-binary LDPC codes

In the present paper, we derive an upper bound on the average network breakdown probability of packet networks with unreliable relay nodes. We here assume that relay nodes get independently broken with a given node breakdown probability. A survivor graph is the induced subgraph obtained by removing the broken relay nodes and their connecting edges from the original graph. If the survivor network is disconnected, we consider a network breakdown happens. The primal contribution of the paper is to derive an upper bound on the average network breakdown probability, where the expectation is taken over a regular graph ensemble. The proof of the bound is based on a natural one-to-one correspondence between a regular graph and a regular bipartite graph, and also on enumeration of bipartite graphs satisfying certain conditions. This proof argument is inspired by the analysis of weight distribution for low-density parity-check codes. Compared with estimates of the average network breakdown probability obtained by computer experiments, it is observed that the upper bound provides the values which are not only upper bounds but also precise estimates of the network breakdown probability when the node breakdown probability is small.

Analysis of breakdown probability of wireless sensor networks with unreliable relay nodes

This paper constructs a non-binary code correcting a single b-burst of insertions or deletions with a large cardinality. This paper also proposes a decoding algorithm of this code and evaluates a lower bound of the cardinality of this code. Moreover, we evaluate an asymptotic upper bound on the cardinality of codes which correct a single burst of insertions or deletions.

An Improvement of Non-binary Code Correcting Single b-Burst of Insertions or Deletions

The fixed points of the belief propagation decoder for non-binary low-density parity-check (LDPC) codes are referred to as stopping constellations. In this paper, we give the stopping constellation distributions for the irregular non-binary LDPC code ensembles defined over the general linear group. Moreover, we derive the exponential growth rate of the average number of the stopping constellation distributions in the limit of large code length.

Takayuki Nozaki

Papers

Bounded Single Insertion/Deletion Correcting Codes

Error floors of non-binary LDPC codes

Analysis of breakdown probability of wireless sensor networks with unreliable relay nodes

An Improvement of Non-binary Code Correcting Single b-Burst of Insertions or Deletions

Analysis of stopping constellation distribution for irregular non-binary LDPC code ensemble