Concentration of Measure Inequalities in Information Theory, Communications and Coding (Second Edition)

During the last two decades, concentration inequalities have been the subject of exciting developments in various areas, including convex geometry, functional analysis, statistical physics, high-dimensional statistics, pure and applied probability theory, information theory, theoretical computer science, and learning theory. 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, as well 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. The basic ingredients of the entropy method are discussed first in the context of logarithmic Sobolev inequalities, which underlie the so-called functional approach to concentration of measure, and then from a complementary information-theoretic viewpoint based on transportation-cost inequalities and probability in metric spaces. 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 to problems in communications and coding, including strong converses, empirical distributions of good channel codes, and an information-theoretic converse for concentration of measure.Comment: Foundations and Trends in Communications and Information Theory, vol. 10, no 1-2, pp. 1-248, 2013. Second edition was published in October 2014. ISBN to printed book: 978-1-60198-906-

Classes of codes from quadratic surfaces of PG(3,q)

We examine classes of binary linear error correcting codes constructed from certain sets of lines defined relative to one of the two classical quadratic surfaces in $PG(3,q)$. We give an overview of some of the properties of the codes, providing proofs where the results are new. In particular, we use geometric techniques to find small weight codewords, and hence, bound the minimum distance

Constructions and Noise Threshold of Hyperbolic Surface Codes

We show how to obtain concrete constructions of homological quantum codes based on tilings of 2D surfaces with constant negative curvature (hyperbolic surfaces). This construction results in two-dimensional quantum codes whose tradeoff of encoding rate versus protection is more favorable than for the surface code. These surface codes would require variable length connections between qubits, as determined by the hyperbolic geometry. We provide numerical estimates of the value of the noise threshold and logical error probability of these codes against independent X or Z noise, assuming noise-free error correction

Balanced Product Quantum Codes

This work provides the first explicit and non-random family of $[[N,K,D]]$ LDPC quantum codes which encode $K \in \Theta(N^\frac{4}{5})$ logical qubits with distance $D \in \Omega(N^\frac{3}{5})$. The family is constructed by amalgamating classical codes and Ramanujan graphs via an operation called balanced product. Recently, Hastings-Haah-O'Donnell and Panteleev-Kalachev were the first to show that there exist families of LDPC quantum codes which break the $\operatorname{polylog}(N)\sqrt{N}$ distance barrier. However, their constructions are based on probabilistic arguments which only guarantee the code parameters with high probability whereas our bounds hold unconditionally. Further, balanced products allow for non-abelian twisting of the check matrices, leading to a construction of LDPC quantum codes that can be shown to have $K\in \Theta(N)$ and that we conjecture to have linear distance $D\in \Theta(N)$.Comment: 23 pages, 11 figure