5 research outputs found
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
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
Constructions for Hadamard matrices, Clifford algebras,and their relation to amicability / anti-amicability graphs
It is known that the Williamson construction for Hadamard matrices can be generalized to constructions using sums of tensor products. This pa-per describes a specific construction using real monomial representations of Of Clifford algebras, and its conne
Asymptotic existence of orthogonal designs
v, 115 leaves ; 29 cmAn orthogonal design of order n and type (si,..., se), denoted OD(n; si,..., se), is a square matrix X of order n with entries from {0, ±x1,..., ±xe}, where the Xj’s are commuting variables, that satisfies XX* = ^ ^g=1 sjx^j In, where X* denotes the transpose of X, and In is the identity matrix of order n.
An asymptotic existence of orthogonal designs is shown. More precisely, for any Atuple (s1,..., se) of positive integers, there exists an integer N = N(s1,..., se) such that for each n > N, there is an OD(2n(s1 + ... + se); 2ns1,..., 2nse). This result of Chapter 5 complements a result of Peter Eades et al. which in turn implies that if the positive integers s1, s2,..., se are all highly divisible by 2, then there is a full orthogonal design of type (s1, s2,..., se).
Some new classes of orthogonal designs related to weighing matrices are obtained in Chapter 3.
In Chapter 4, we deal with product designs and amicable orthogonal designs, and a construction method is presented.
Signed group orthogonal designs, a natural extension of orthogonal designs, are introduced in Chapter 6. Furthermore, an asymptotic existence of signed group orthogonal designs is obtained and applied to show the asymptotic existence of orthogonal designs