Orthogonal Designs IV: Existence Questions

@article{Geramita1975OrthogonalDI,
  title={Orthogonal Designs IV: Existence Questions},
  author={Anthony V. Geramita and Jennifer Seberry},
  journal={J. Comb. Theory, Ser. A},
  year={1975},
  volume={19},
  pages={66-83}
}
In [5] Raghavarao showed that if n = 2 (mod 4) and A is a {O, 1, -1} matrix satisfying AAt = (n 1) In. then n 1 = a2 b2 for a, b integers. In [4] van Lint and Seidel giving a proof modeled on a proof of the Witt cancellation theorem, proved more generally that if n is as above and A is a rational matrix satisfying AAt = kIn then k = q12 + q22 (q1, q2 E Q, the rational numbers). Consequently, if k is an integer then k = a2 + b2 for two integers a and b. In [1] we showed that if, in addition, A… Expand

Tables and Topics from this paper

A note on orthogonal designs in order eighty
This is a short note showing the existence of all twovariable designs in order 80 except possibly (13, 64) and (15, 62) which have not yet been construced. The designs are constructed using designsExpand
Orthogonal designs V: Orders divisible by eight
Constructions are given for orthogonal designs in orders divisible by eight. These are then used to show all two variable orthogonal designs exist in orders 24, 32 and 48. The existence of twoExpand
The skew - weighing matrix conjecture
We review the history of the skew-weighing matrix conjecture and show that there exist skew-symmetric weighing matrices W (21.2t, k) for all k=0,l,.....,21.2t I, t ≥ 4 a positive integer. Hence thereExpand
Some results on weighing matrices
It is shown that if q is a prime power then there exists a circulant weighing matrix of order q 2 + q + 1 with q 2 nonzero elements per row and column. This result allows the bound N to be lowered inExpand
New weighing matrices and orthogonal designs constructed using two sequences with zero autocorrelation function – a review
Abstract The book, Orthogonal Designs : Quadratic Forms and Hadamard Matrices , Marcel Dekker, New York-Basel, 1979, by A.V. Geramita and Jennifer Seberry, has now been out of print for almost twoExpand
On sufficient conditions for some orthogonal designs and sequences with zero autocorrelation function
We give new sets of sequences with entries from {0, ±a, ±b, ±c, ±d} on the commuting variables a, b, c, d and zero autocorrelation function. Then we use these sequences to construct some newExpand
On weighing matrices
We give new sets of {0, 1, -1} sequences with zero autocorrelation function, new constructions for weighing matrices and review the weighing matrix conjecture for orders 4t, t є {1,...,25}Expand
A Journey of Discovery : Orthogonal Matrices and Wireless Communications
Real orthogonal designs were first introduced in the 1970’s, followed shortly by the introduction of complex orthogonal designs. These designs can be described simply as square matrices whose columnsExpand
A survey of orthogonal designs
This paper surveys orthogonal designs which are an overview of Baumert-Hall arrays, Hadamard matrices and weighing matrices. The known results are given and unsolved problems indicated. DisciplinesExpand
Some new weighing matrices using sequences with zero autocorrelation function
TLDR
The skew weighing matrix conjecture for orders 2t is verified, proving the conjecture for t ≥ 3.13, t ≥ 5, and new results for 2t.15 are given. Expand
...
1
2
...

References

SHOWING 1-8 OF 8 REFERENCES
Orthogonal (0,1,-1) matrices
We study the conjecture: There exists a square (0,l,-l)-matrix W = W(w,k) of order w satisfying WWT= kIw for all k = 0, 1,..., w when w = 0 (mod 4). We prove the conjecture is true for 4, 8, 12, 16,Expand
Orthogonal designs II
AbstractOrthogonal designs are a natural generalization of the Baumert-Hall arrays which have been used to construct Hadamard matrices. We continue our investigation of these designs and show thatExpand
Orthogonal designs III: Weighing matrices
A weighing matrix W = W(n,k) of order n and weight k is a square (0,l,-l)-matrix satisfying WWt -kIn An orthogonal design of order n on a single variable is a weighing matrix and consequently theExpand
Equilateral point sets in elliptic geometry
This chapter highlights equilateral point sets in elliptic geometry. Elliptic space of r−1 dimensions E r−1 is obtained from r -dimensional vector space R r with inner product ( a , b ). For 1 , anyExpand
Combinatorics: room squares, sum-free sets, Hadamard matrices
Now welcome, the most inspiring book today from a very professional writer in the world, combinatorics room squares sum free sets hadamard matrices. This is the book that many people in the worldExpand
Integral Matrices,
  • Pure and Applied Mathematics,
  • 1972