Constructions of complex Hadamard matrices via tiling Abelian groups

Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent very general construction of complex Hadamard matrices due to Dita via a natural tiling construction. Then we find some necessary conditions for any given complex Hadamard matrix to be equivalent to a Dita-type matrix. Finally, using another tiling construction, due to Szabo, we arrive at new parametric families of complex Hadamard matrices of order 8, 12 and 16, and we use our necessary conditions to prove that these families do not arise with Dita's construction. These new families complement the recent catalogue of complex Hadamard matrices of small order.Comment: 15 page

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

Unitary Representations of Wavelet Groups and Encoding of Iterated Function Systems in Solenoids

For points in $d$ real dimensions, we introduce a geometry for general digit sets. We introduce a positional number system where the basis for our representation is a fixed $d$ by $d$ matrix over \bz. Our starting point is a given pair $(A, \mathcal D)$ with the matrix $A$ assumed expansive, and $\mathcal D$ a chosen complete digit set, i.e., in bijective correspondence with the points in \bz^d/A^T\bz^d. We give an explicit geometric representation and encoding with infinite words in letters from $\mathcal D$. We show that the attractor $X(A^T,\mathcal D)$ for an affine Iterated Function System (IFS) based on $(A,\mathcal D)$ is a set of fractions for our digital representation of points in \br^d. Moreover our positional "number representation" is spelled out in the form of an explicit IFS-encoding of a compact solenoid \sa associated with the pair $(A,\mathcal D)$. The intricate part (Theorem \ref{thenccycl}) is played by the cycles in \bz^d for the initial $(A,\mathcal D)$-IFS. Using these cycles we are able to write down formulas for the two maps which do the encoding as well as the decoding in our positional $\mathcal D$-representation. We show how some wavelet representations can be realized on the solenoid, and on symbolic spaces

Parametrizing Complex Hadamard Matrices

The purpose of this paper is to introduce new parametric families of complex Hadamard matrices in two different ways. First, we prove that every real Hadamard matrix of order N>=4 admits an affine orbit. This settles a recent open problem of Tadej and Zyczkowski, who asked whether a real Hadamard matrix can be isolated among complex ones. In particular, we apply our construction to the only (up to equivalence) real Hadamard matrix of order 12 and show that the arising affine family is different from all previously known examples. Second, we recall a well-known construction related to real conference matrices, and show how to introduce an affine parameter in the arising complex Hadamard matrices. This leads to new parametric families of orders 10 and 14. An interesting feature of both of our constructions is that the arising families cannot be obtained via Dita's general method. Our results extend the recent catalogue of complex Hadamard matrices, and may lead to direct applications in quantum-information theory.Comment: 16 pages; Final version. Submitted to: European Journal of Combinatoric