Complex Hadamard matrices and equiangular tight frames

In this paper, we give a new construction of parametric families of complex Hadamard matrices of square orders, and connect them to equiangular tight frames. The results presented here generalize some of the recent ideas of Bodmann et al. [3] and extend the list of known equiangular tight frames. In particular, a (36, 21)-frame coming from a nontrivial cube root signature matrix is obtained for the first time.     

Embedding matrices in Hadamard matrices

It is shown that if A is any n×n matrix of zeros and ones, and if k is the smallest number not less than n which is the order of an Hadamard matrix, then A is a submatrix of an Hadamard matrix of order k².     

Deciding Hadamard equivalence of Hadamard matrices

Equivalence of Hadamard matrices can be decided inO(log² n) space, and hence in subexponential time. These resource bounds follow from the existence of small distinguishing sets.     

Generalized Hadamard matrices whose transposes are not generalized Hadamard matrices

In answer to “Research Problem 16” in Horadam's recent book Hadamard matrices and their applications, we provide a construction for generalized Hadamard matrices whose transposes are not generalized Hadamard matrices. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 456–458, 2009     

Power Hadamard matrices

We introduce power Hadamard matrices, in order to study the structure of (group) generalized Hadamard matrices, Butson (generalized) Hadamard matrices and other related orthogonal matrices, with which they share certain common characteristics. The new objects turn out to be as interesting, and perhaps as useful, as the objects that motivated them.We develop a basic theory of power Hadamard matrices, explore these relationships, and offer some new insights into old results. For example, we show that all 4×4 Butson Hadamard matrices are equivalent to circulant ones, and how to move between equivalence classes.We provide, among other new things, an infinite family of circulant Butson Hadamard matrices that extends a known class to include one of each positive integer order.Dedication: In 1974 Jennifer Seberry (Wallis) introduced what was then a totally new structure, orthogonal designs, in order to study the existence and construction of Hadamard matrices. They have proved their worth for this purpose, and have also become an object of interest for their own sake and in applications (e.g., [H.J.V. Tarok, A.R. Calderbank, Space-time block codes from orthogonal designs, IEEE Trans. Inf. Theory 45 (1999) 1456-1467. [26]]). Since then many other generalizations of Hadamard matrices have been introduced, including some discussed herein. In the same spirit we introduce a new object showing this kind of promise.Seberry's contributions to this field are not limited to her own work, of which orthogonal designs are but one example—she has mentored many young mathematicians who have expanded her legacy by making their own marks in this field. It is fitting, therefore, that our contribution to this volume is a collaboration between one who has worked in this field for over a decade and an undergraduate student who had just completed his third year of study at the time of the work.     

Rectangular Hadamard matrices

A method for the construction of v × b matrices with elements 1, −1, such that XX′ = bI, is given.     

Some New Hadamard matrices

Dicyclic solutions to BIBDs with parameters (2t + 1, 4t + 2, 2t, t, t – 1) can be embedded in a Hadamard matrix of order 4t + 4. New Hadamard matrices of order 44 are constructed using this embedding.     

Balancedly splittable Hadamard matrices

Balancedly splittable Hadamard matrices are introduced and studied. A connection is made to the Hadamard diagonalizable strongly regular graphs, maximal equiangular lines set, and unbiased Hadamard matrices. Several construction methods are presented. As an application, commutative association schemes of 4, 5, and 6 classes are constructed.     

Decomposition of Hadamard matrices

The decomposition is defined. The components are each as an orthogonal matrix with elements 0, ±1. In pairs they satisfy XY^T + YX^T = 0. It is conjectured that every Hadamard matrix of order mn is decomposable into m components for m = 4 or 8.     

Hadamard products of matrices

The entry-wise product of arbitrary n × ncomplex matrices is studied. The principal tools used include the Kionecker product, field of values and diagonal multiplications. Inclusion theorems for the field of values and spectrum are developed in the general case and refined in special cases. These are employed to obtain inequalities involving the real parts of the characteristic roots and the numerical radius, and previously known results are found to be special cases of several of the theorems. In addition, the case of positive stable matrices is considered and a new class of nonnegative stable matrices is introduced, studied and related to D-stability.     

Circulant partial Hadamard matrices

A question arising in stream cypher cryptanalysis is reframed and generalized in the setting of Hadamard matrices as follows: For given n, what is the maximum value of k   for which there exists a k×n$k\times n$(±1)$(\pm 1)$-matrix A   such that AA^T=n_Ik$A{A}^{\mathsf{T}}=n{I}_k$, with each row after the first obtained by a cyclic shift of its predecessor by one position? For obvious reasons we call such matrices circulant partial Hadamard matrices. Further, what is the maximum value of k subject to the condition that the row sums are equal to r?     

Equivalence of Hadamard matrices

Supposem is a square-free odd integer, andA andB are any two Hadamard matrices of order 4m. We will show thatA andB are equivalent over the integers (that is,B can be obtained fromA using elementary row and column operations which involve only integers).     

Cocyclic two-circulant core Hadamard matrices

Journal of Algebraic Combinatorics - The two-circulant core (TCC) construction for Hadamard matrices uses two sequences with almost perfect autocorrelation to construct a Hadamard matrix. A...     

Equivalence of Butson-type Hadamard matrices

Journal of Algebraic Combinatorics - Two matrices $$H_1$$ and $$H_2$$ with entries from a multiplicative group G are said to be monomially equivalent, denoted by $$H_1cong H_2$$ , if one of the...     

Hadamard matrices and difference families

Translated from Matematicheskie Zametki, Vol. 47, No. 3, pp. 11–16, March, 1990.     

A construction for Hadamard matrices

Let 2ⁿm be the order of an Hadamard matrix. Using block Golay sequences, a class of Hadamard matrices of order (r+4ⁿ+1)4ⁿ⁺¹m² is constructed, where r is the length of a Golay sequence.     

Hadamard matrices and projective planes

The construction of a Hadamard matrix of order n² from a projective plane of order n, n even, is given. Alternative constructions, specialized to the case n = 10, from sets of mutually orthogonal Latin squares are also given. Special properties of the Hadamard matrices are discussed and a partial example is given in the case n = 10.     

Orbit matrices of Hadamard matrices and related codes

In this paper we introduce the notion of orbit matrices of Hadamard matrices with respect to their permutation automorphism groups and show that under certain conditions these orbit matrices yield self-orthogonal codes. As a case study, we construct codes from orbit matrices of some Paley type I and Paley type II Hadamard matrices. In addition, we construct four new symmetric (100,45,20) designs which correspond to regular Hadamard matrices, and construct codes from their orbit matrices. The codes constructed include optimal, near-optimal self-orthogonal and self-dual codes, over finite fields and over ${\mathbb{Z}}_4$.