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

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

Tremain equiangular tight frames

Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph.Comment: 11 page

Matrix constructions of divisible designs

AbstractWe present two new constructions of group divisible designs. We use skew-symmetric Hadamard matrices and certain strongly regular graphs together with (v, k, λ)-designs. We include many examples, in particular several new series of divisible difference sets

Hadamard Tensors and Lower Bounds on Multiparty Communication Complexity

We develop a new method for estimating the discrepancy of tensors associated with multiparty communication problems in the ``Number on the Forehead\u27\u27 model of Chandra, Furst and Lipton. We define an analogue of the Hadamard property of matrices for tensors in multiple dimensions and show that any $k$-party communication problem represented by a Hadamard tensor must have $Omega(n/2^k)$ multiparty communication complexity. We also exhibit constructions of Hadamard tensors, giving $Omega(n/2^k)$ lower bounds on multiparty communication complexity for a new class of explicitly defined Boolean functions