scholarly journals Amicable orthogonal designs

1976 ◽  
Vol 14 (2) ◽  
pp. 303-314 ◽  
Author(s):  
Peter J. Robinson
Keyword(s):  
Hadamard Matrices ◽  
Orthogonal Designs ◽  
Image Position

A powerful tool in the construction of orthogonal designs has been amicable orthogonal designs. Recent results in the construction of Hadamard matrices has led to the need to find amicable orthogonal designs A, B in order n and of types (u1, U2, …, u6) and (ν1, ν2, …, νr) respectively satisfying At = -A, Bt = B, and ABt = BAt withFor simplicity, we say A, B are amicable orthogonal designs of type (u1, u2, …, us; v1, v2, …, vr).We completely answer the question in order 8 by showing (1, 2, 2, 2; 8), (1, 2, 4; 2, 2, 4), (2, 2, 3; 2, 6), (7, 1, 7) and those designs derived from the above are the only possible.We use our results to obtain new orthogonal designs in order 32.

scholarly journals Short Amicable sets and Kharaghani type orthogonal designs

2001 ◽  
Vol 64 (3) ◽  
pp. 495-504 ◽  
Author(s):  
Christos Koukouvinos ◽  
Jennifer Seberry
Keyword(s):  
Hadamard Matrices ◽  
Weighing Matrices ◽  
Orthogonal Designs

Dedicated to Professor George SzekeresShort amicable sets were introduced recently and have many applications. The construction of short amicable sets has lead to the construction of many orthogonal designs, weighing matrices and Hadamard matrices. In this paper we give some constructions for short amicable sets as well as some multiplication theorems. We also present a table of the short amicable sets known to exist and we construct some infinite families of short amicable sets and orthogonal designs.

Amicable Orthogonal Designs-Existence

1976 ◽  
Vol 28 (5) ◽  
pp. 1006-1020 ◽  
Author(s):  
Warren Wolfe
Keyword(s):  
Orthogonal Design ◽  
Orthogonal Designs ◽  
Image Position

Definition. An orthogonal design in order n and of type (u1, … , us) on the commuting variables x1, . . . , xs is an n X n matrix, X, with entries from the set ﹛0, ±x1, … , ±xs﹜ such that

scholarly journals Group properties of Hadamard matrices

1976 ◽  
Vol 21 (2) ◽  
pp. 247-256 ◽  
Author(s):  
Marshall Hall
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Symmetric Design ◽  
Image Position ◽  
Square Matrix

An Hadamard matrix H is a square matrix of order n all of whose entries are ± 1 such thatThere are matrices of order 1 and 2and for all other Hadamard matrices the order n is a multiple of 4, n = 4m. It is a reasonable conjecture that Hadamard matrices exist for every order which is a multiple of 4 and the lowest order in doubt is 268. With every Hadamard matrix H4m a symmetric design D exists with

Semi-automorphisms of Hadamard matrices

1975 ◽  
Vol 77 (3) ◽  
pp. 459-473 ◽  
Author(s):  
Marshall Hall
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Image Position

A Hadamard matrix Hn is an n by n matrix H = [hij], i, j = 1, …, n in which every entry hij is + 1 or − 1, such thatIt is well known that possible orders are n = 1, 2 and n = 4m. An automorphism α of H is given by a pair P, Q of monomial ± 1 matrices such thatHere P permutes and changes signs of rows, while Q acts similarly on columns.

scholarly journals Construction of generalized rotations and quasi-orthogonal matrices

2019 ◽  
Vol 7 (1) ◽  
pp. 107-113
Author(s):  
Luis Verde-Star
Keyword(s):  
Image Processing ◽  
Hadamard Matrices ◽  
Complex Numbers ◽  
Data Communications ◽  
Matrix Representations ◽  
Orthogonal Matrices ◽  
Orthogonal Designs ◽  
The Matrix ◽  
The One

Abstract We propose some methods for the construction of large quasi-orthogonal matrices and generalized rotations that may be used in applications in data communications and image processing. We use certain combinations of constructions by blocks similar to the one used by Sylvester to build Hadamard matrices. The orthogonal designs related with the matrix representations of the complex numbers, the quaternions, and the octonions are used in our construction procedures.

scholarly journals (v, k, λ) Configurations and Hadamard matrices

1970 ◽  
Vol 11 (3) ◽  
pp. 297-309 ◽  
Author(s):  
Jennifer Wallis
Keyword(s):  
Hadamard Matrix ◽  
Hadamard Matrices ◽  
Image Position

Using the terminology in 2 (where the expression m-type is also explained) we will prove the following theorems: Theorem 1. If there exist (i) a skew-Hadamard matrix H = U+I of order h, (ii)m-type matrices M = W+I and N = NT of order m, (iii) three matrices X, Y, Z of order x = 3 (mod 4) satisfying (a) XYT, YZT and ZXT all symmetric, and (b) XXT = aIx+bJxthen is an Hadamard matrix of order mxh.

scholarly journals Hadamard Matrices, Orthogonal Designs and Construction Algorithms

Designs 2002 ◽  
2003 ◽  
pp. 133-205 ◽  
Author(s):  
Stelios Georgiou ◽  
Christos Koukouvinos ◽  
Jennifer Seberry
Keyword(s):  
Hadamard Matrices ◽  
Orthogonal Designs ◽  
Construction Algorithms

Using product designs to construct orthogonal designs

1977 ◽  
Vol 16 (2) ◽  
pp. 297-305 ◽  
Author(s):  
Peter J. Robinson
Keyword(s):  
Orthogonal Design ◽  
Orthogonal Designs ◽  
Image Position

This paper produces new types of designs, called product designs, which prove extremely useful for constructing orthogonal designs. An orthogonal design of order 2t and typeis constructed. This design often meets the Radon bound for the number of variables.We also show that all orthogonal designs of order 2t and type (a, b, c, d, 2t-a-b-c-d), with 0 < a + b + c + d < 2t, exist for t = 5, 6, and 7.

Limits on Pairwise Amicable Orthogonal Designs

1981 ◽  
Vol 33 (5) ◽  
pp. 1043-1054
Author(s):  
Warren Wolfe
Keyword(s):  
Orthogonal Design ◽  
Orthogonal Matrices ◽  
Orthogonal Designs ◽  
Image Position

Anorthogonal design in order n of type(u1, …,ut) on the commuting variablesx1, …,xtis ann×nmatrixXwith entries 0, ±x1, …, ±xtsuch thatIn [5] Geramita and Wallis show that ifn= 24a+b·n0, wheren0is odd and 0 ≦b> 4, thent≧ρ(n)= 8a+ 2b. The result is essentially Radon's limit on the number of anti-commuting, real, anti-symmetric, orthogonal matrices in ordern. Garamita and Pullman show that this limit is sharp for orthogonal designs: i.e., givenn, there exists an orthogonal design in ordernwithρ(n)variables [6].Two orthogonal designs,XandF, are calledamicableifXYt=YXt.Such pairs of orthogonal designs are especially useful in generating new orthogonal designs [5] or [6].

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