108 research outputs found
Amicable T-matrices and applications
iii, 49 leaves ; 29 cmOur main aim in this thesis is to produce new T-matrices from the set of existing
T-matrices. In Theorem 4.3 a multiplication method is introduced to generate new
T-matrices of order st, provided that there are some specially structured T-matrices
of orders s and t. A class of properly amicable and double disjoint T-matrices are
introduced. A number of properly amicable T-matrices are constructed which includes
2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 18, 22.
To keep the new matrices disjoint an extra condition is imposed on one set of
T-matrices and named double disjoint T-matrices. It is shown that there are some
T-matrices that are both double disjoint and properly amicable. Using these matrices
an infinite family of new T-matrices are constructed.
We then turn our attention to the application of T-matrices to construct orthogonal
designs and complex Hadamard matrices.
Using T-matrices some orthogonal designs constructed from 16 circulant matrices
are constructed. It is known that having T-matrices of order t and orthogonal designs
constructible from 16 circulant matrices lead to an infinite family of orthogonal designs.
Using amicable T-matrices some complex Hadamard matrices are shown to exist
Some Constructions for Amicable Orthogonal Designs
Hadamard matrices, orthogonal designs and amicable orthogonal designs have a
number of applications in coding theory, cryptography, wireless network
communication and so on. Product designs were introduced by Robinson in order
to construct orthogonal designs especially full orthogonal designs (no zero
entries) with maximum number of variables for some orders. He constructed
product designs of orders , and and types and ,
respectively. In this paper, we first show that there does not exist any
product design of order , , and type where the notation is used to show that repeats
times. Then, following the Holzmann and Kharaghani's methods, we construct some
classes of disjoint and some classes of full amicable orthogonal designs, and
we obtain an infinite class of full amicable orthogonal designs. Moreover, a
full amicable orthogonal design of order and type is constructed.Comment: 12 pages, To appear in the Australasian Journal of Combinatoric
Constructions for orthogonal designs using signed group orthogonal designs
Craigen introduced and studied signed group Hadamard matrices extensively and
eventually provided an asymptotic existence result for Hadamard matrices.
Following his lead, Ghaderpour introduced signed group orthogonal designs and
showed an asymptotic existence result for orthogonal designs and consequently
Hadamard matrices. In this paper, we construct some interesting families of
orthogonal designs using signed group orthogonal designs to show the capability
of signed group orthogonal designs in generation of different types of
orthogonal designs.Comment: To appear in Discrete Mathematics (Elsevier). No figure
Twin bent functions, strongly regular Cayley graphs, and Hurwitz-Radon theory
The real monomial representations of Clifford algebras give rise to two
sequences of bent functions. For each of these sequences, the corresponding
Cayley graphs are strongly regular graphs, and the corresponding sequences of
strongly regular graph parameters coincide. Even so, the corresponding graphs
in the two sequences are not isomorphic, except in the first 3 cases. The proof
of this non-isomorphism is a simple consequence of a theorem of Radon.Comment: 13 pages. Addressed one reviewer's questions in the Discussion
section, including more references. Resubmitted to JACODES Math, with updated
affiliation (I am now an Honorary Fellow of the University of Melbourne
Small orders of Hadamard matrices and base sequences
We update the list of odd integers n<10000 for which an Hadamard matrix of
order 4n is known to exist. We also exhibit the first example of base sequences
BS(40,39). Consequently, there exist T-sequences TS(n) of length n=79. The
first undecided case has the length n=97.Comment: 7 page
Twin bent functions and Clifford algebras
This paper examines a pair of bent functions on and their
relationship to a necessary condition for the existence of an automorphism of
an edge-coloured graph whose colours are defined by the properties of a
canonical basis for the real representation of the Clifford algebra
Some other necessary conditions are also briefly examined.Comment: 11 pages. Preprint edited so that theorem numbers, etc. match those
in the published book chapter. Final post-submission paragraph added to
Section 6. in "Algebraic Design Theory and Hadamard Matrices: ADTHM,
Lethbridge, Alberta, Canada, July 2014", Charles J. Colbourn (editor), pp.
189-199, 201
Supplementary difference sets with symmetry for Hadamard matrices
First we give an overview of the known supplementary difference sets (SDS)
(A_i), i=1..4, with parameters (n;k_i;d), where k_i=|A_i| and each A_i is
either symmetric or skew and k_1 + ... + k_4 = n + d. Five new Williamson
matrices over the elementary abelian groups of order 25, 27 and 49 are
constructed. New examples of skew Hadamard matrices of order 4n for n=47,61,127
are presented. The last of these is obtained from a (127,57,76)-difference
family that we have constructed. An old non-published example of G-matrices of
order 37 is also included.Comment: 16 pages, 2 tables. A few minor changes are made. The paper will
appear in Operators and Matrice
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