4,486 research outputs found
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
Some non-existence and asymptotic existence results for weighing matrices
Orthogonal designs and weighing matrices have many applications in areas such
as coding theory, cryptography, wireless networking and communication. In this
paper, we first show that if positive integer cannot be written as the sum
of three integer squares, then there does not exist any skew-symmetric weighing
matrix of order and weight , where is an odd positive integer. Then
we show that for any square , there is an integer such that for each
, there is a symmetric weighing matrix of order and weight .
Moreover, we improve some of the asymptotic existence results for weighing
matrices obtained by Eades, Geramita and Seberry.Comment: To appear in International Journal of Combinatorics (Hindawi). in
Int. J. Combin. (Feb 2016
Quantum Algorithms for Weighing Matrices and Quadratic Residues
In this article we investigate how we can employ the structure of
combinatorial objects like Hadamard matrices and weighing matrices to device
new quantum algorithms. We show how the properties of a weighing matrix can be
used to construct a problem for which the quantum query complexity is
ignificantly lower than the classical one. It is pointed out that this scheme
captures both Bernstein & Vazirani's inner-product protocol, as well as
Grover's search algorithm.
In the second part of the article we consider Paley's construction of
Hadamard matrices, which relies on the properties of quadratic characters over
finite fields. We design a query problem that uses the Legendre symbol chi
(which indicates if an element of a finite field F_q is a quadratic residue or
not). It is shown how for a shifted Legendre function f_s(i)=chi(i+s), the
unknown s in F_q can be obtained exactly with only two quantum calls to f_s.
This is in sharp contrast with the observation that any classical,
probabilistic procedure requires more than log(q) + log((1-e)/2) queries to
solve the same problem.Comment: 18 pages, no figures, LaTeX2e, uses packages {amssymb,amsmath};
classical upper bounds added, presentation improve
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
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