108 research outputs found

    Amicable T-matrices and applications

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    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

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    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 44, 88 and 1212 and types (1(3);1(3);1),\big(1_{(3)}; 1_{(3)}; 1\big), (1(3);1(3);5)\big(1_{(3)}; 1_{(3)}; 5\big) and (1(3);1(3);9)\big(1_{(3)}; 1_{(3)}; 9\big), respectively. In this paper, we first show that there does not exist any product design of order n≠4n\neq 4, 88, 1212 and type (1(3);1(3);n−3),\big(1_{(3)}; 1_{(3)}; n-3\big), where the notation u(k)u_{(k)} is used to show that uu repeats kk 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 292^9 and type (2(8)6;2(8)6)\big(2^6_{(8)}; 2^6_{(8)}\big) is constructed.Comment: 12 pages, To appear in the Australasian Journal of Combinatoric

    Constructions for orthogonal designs using signed group orthogonal designs

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    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

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    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

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    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

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    This paper examines a pair of bent functions on Z22m\mathbb{Z}_2^{2m} 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 Rm,m.\mathbb{R}_{m,m}. 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

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    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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