R. Calderbank

Bio: R. Calderbank is an academic researcher from Bell Labs. The author has an hindex of 1, co-authored 1 publications receiving 525 citations.

The Geometry of Two-Weight Codes

Abstract: On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles (n,k,h 1 h 2 ) projectifs et certains graphes fortement reguliers

Spectra of graphs

Abstract: This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

Quantum error correction via codes over GF(4)

Abstract: The unreasonable effectiveness of quantum computing is founded on coherent quantum superposition or entanglement which allows a large number of calculations to be performed simultaneously. This coherence is lost as a quantum system interacts with its environment. In the present paper the problem of finding quantum-error-correcting codes is transformed into one of finding additive codes over the field GF(4) which are self-orthogonal with respect to a certain trace inner product. Many new codes and new bounds are presented, as well as a table of upper and lower bounds on such codes of length up to 30 qubits.

Algebraic Geometric Codes: Basic Notions

Abstract: The book is devoted to the theory of algebraic geometric codes, a subject formed on the border of several domains of mathematics On one side there are such classical areas as algebraic geometry and number theory; on the other, information transmission theory, combinatorics, finite geometries, dense packings, etc The authors give a unique perspective on the subject Whereas most books on coding theory build up coding theory from within, starting from elementary concepts and almost always finishing without reaching a certain depth, this book constantly looks for interpretations that connect coding theory to algebraic geometry and number theory There are no prerequisites other than a standard algebra graduate course The first two chapters of the book can serve as an introduction to coding theory and algebraic geometry respectively Special attention is given to the geometry of curves over finite fields in the third chapter Finally, in the last chapter the authors explain relations between all of these: the theory of algebraic geometric codes

A Class of Two-Weight and Three-Weight Codes and Their Applications in Secret Sharing

Abstract: In this paper, a class of two-weight and three-weight linear codes over $ {\mathrm {GF}}(p)$ is constructed, and their application in secret sharing is investigated. Some of the linear codes obtained are optimal in the sense that they meet certain bounds on linear codes. These codes have applications also in authentication codes, association schemes, and strongly regular graphs, in addition to their applications in consumer electronics, communication and data storage systems.

Linear Codes From Some 2-Designs

Abstract: A classical method of constructing a linear code over $ {\mathrm {GF}}(q)$ with a $t$ -design is to use the incidence matrix of the $t$ -design as a generator matrix over $ {\mathrm {GF}}(q)$ of the code. This approach has been extensively investigated in the literature. In this paper, a different method of constructing linear codes using specific classes of 2-designs is studied, and linear codes with a few weights are obtained from almost difference sets, difference sets, and a type of 2-designs associated to semibent functions. Two families of the codes obtained in this paper are optimal. The linear codes presented in this paper have applications in secret sharing and authentication schemes, in addition to their applications in consumer electronics, communication and data storage systems. A coding-theory approach to the characterization of highly nonlinear Boolean functions is presented.