A Generalised Hadamard Transform

A Generalised Hadamard Transform for multi-phase or multilevel signals is introduced, which includes the Fourier, Generalised, Discrete Fourier, Walsh-Hadamard and Reverse Jacket Transforms. The jacket construction is formalised and shown to admit a tensor product decomposition. Primary matrices under this decomposition are identified. New examples of primary jacket matrices of orders 8 and 12 are presented.Comment: To appear in the proceedings of the 2005 IEEE International Symposium on Information Theory, Adelaide, Australia, September 4-9, 200

Isolated Hadamard Matrices from Mutually Unbiased Product Bases

A new construction of complex Hadamard matrices of composite order d=pq, with primes p,q, is presented which is based on pairs of mutually unbiased bases containing only product states. For product dimensions d < 100, we illustrate the method by deriving many previously unknown complex Hadamard matrices. We obtain at least 12 new isolated matrices of Butson type, with orders ranging from 9 to 91.Comment: 21 pages, identical to published versio

Generalized Heisenberg Algebras and Fibonacci Series

We have constructed a Heisenberg-type algebra generated by the Hamiltonian, the step operators and an auxiliar operator. This algebra describes quantum systems having eigenvalues of the Hamiltonian depending on the eigenvalues of the two previous levels. This happens, for example, for systems having the energy spectrum given by Fibonacci sequence. Moreover, the algebraic structure depends on two functions f(x) and g(x). When these two functions are linear we classify, analysing the stability of the fixed points of the functions, the possible representations for this algebra.Comment: 24 pages, 2 figures, subfigure.st

On quaternary complex Hadamard matrices of small orders

One of the main goals of design theory is to classify, characterize and count various combinatorial objects with some prescribed properties. In most cases, however, one quickly encounters a combinatorial explosion and even if the complete enumeration of the objects is possible, there is no apparent way how to study them in details, store them efficiently, or generate a particular one rapidly. In this paper we propose a novel method to deal with these difficulties, and illustrate it by presenting the classification of quaternary complex Hadamard matrices up to order 8. The obtained matrices are members of only a handful of parametric families, and each inequivalent matrix, up to transposition, can be identified through its fingerprint.Comment: 7 page

Hadamard Matrices and Their Applications

In Hadamard Matrices and Their Applications, K. J. Horadam provides the first unified account of cocyclic Hadamard matrices and their applications in signal and data processing. This original work is based on the development of an algebraic link between Hadamard matrices and the cohomology of finite groups that was discovered fifteen years ago. The book translates physical applications into terms a pure mathematician will appreciate, and theoretical structures into ones an applied mathematician, computer scientist, or communications engineer can adapt and use. The first half of the book exp

Switching edges to randomize networks: What goes wrong and how to fix it

The switching model is a well-known random network model that randomizes a network while keeping its degree sequence fixed. The idea behind the switching model is simple: a network is randomized by repeatedly rewiring pairs of edges. In this paper we demonstrate that despite its simple description, and in part due to it, much can go wrong when implementing the switching model. Specifically, we show that the model needs to be implemented carefully, to avoid biased sampling. We propose a precise definition of the switching model which guides its implementation. Furthermore, we argue that we should refer to the switching model with respect to a specific network class, and in fact define a family of switching models. This formalizes previous use of the switching model to randomize networks from numerous network classes. We show that the properties of these models depend on the network class, and in particular that their stationary distributions differ. Hence it is important to take into account which class of networks is being randomized. We derive conditions, and where possible adjust the models, such that sampling is unbiased for the switching model with respect to eight common network classes. These unbiased null-models are important, since common network analysis techniques such as motif finding and community detection rely on them