Faculty of Arts Mathematics Department Master of Literature by Padraig Ó Catháin Hadamard matrices are an important item of study in combinatorial design theory. In this thesis, we explore the theory of cocyclic development of Hadamard matrices in terms of regular group actions on the expanded design. To this end a general theory of both group development and cocyclic development is formulated....

Say that A is a Hadamard factorization of the identity In of size n if A ◦AT = In, where ◦ denotes the Hadamard or entrywise product, and AT denotes the transpose of A. As n = rk(In) = rk(A ◦ AT ) ≤ rk(A)2, it is clear that the rank of any Hadamard factorization of the identity must be at least √ n. Dietzfelbinger et al. [DHS96] raised the question if this bound can be achieved, and showed a bo...

Hadamard speckle contrast reduction (SCR) is considered to be an effective approach to deal with speckle problems in coherent imaging systems. A Hadamard SCR system is divided into two subsystems, which implement phase patterns projection and reflected waves imaging respectively. The performances of both sub-systems are discussed with numerical simulations and linked to certain parameters so as...

This paper proposes an algorithm which can be used to construct strongly regular graphs from Hadamard matrices.A graph is strongly regular if there are integers and such that every two adjacent vertices have common neighbours and every two non adjacent vertices have common neighbors. Proposed method is mainly based on basic matrix manipulations. If the order of the normalized Hadamard matr ix i...

Hadamard conjugation can be used in the reconstruction of evolutionary trees and analysis of molecular sequence evolution. Despite the number of advantages Hadamard conjugation provides the number of evolutionary substitution models that can be used with this technique is limited. In this paper, we consider the question of whether Hadamard conjugation is limited to group-based evolutionary mode...

Skew Hadamard designs (4n−1, 2n−1, n−1) are associated to order 4n skew Hadamard matrices in the natural way. We study the codes spanned by their incidence matrices A and by I +A and show that they are self-dual after extension (resp. extension and augmentation) over fields of characteristic dividing n. Quadratic Residues codes are obtained in the case of the Paley matrix. Results on the p−rank...

Computational Algebra methods have been used successfully in various problems in many fields of Mathematics. Computational Algebra encompasses a set of powerful algorithms for studying ideals in polynomial rings and solving systems of nonlinear polynomial equations efficiently. The theory of Gröbner bases is a cornerstone of Computational Algebra, since it provides us with a constructive way of...

Automorphism groupS of Hadamard matrices are related to automorphism groups of de.Jigns, and the automorphism groups of the Paley-Hadamard matrices are determined .

A novel method to obtain parameterizations of complex inverse orthogonal matrices is provided. These matrices are natural generalizations of complex Hadamard matrices which depend on non zero complex parameters. The method we use is via doubling the size of inverse complex conference matrices. When the free parameters take values on the unit circle the inverse orthogonal matrices transform into...

A recently proposed matrix factorization for aHadamard matrix of order twelve is shown to be invalid in that thefactored matrix is not Hadamard.