ORTHOGONAL MATRICES WITH ZERO DIAGONAL

@article{Goethals1967ORTHOGONALMW,
  title={ORTHOGONAL MATRICES WITH ZERO DIAGONAL},
  author={Jean-Marie Goethals and J. J. Seidel},
  journal={Canadian Journal of Mathematics},
  year={1967},
  volume={19},
  pages={1001-1010}
}
1. Introduction. C-matrices appear in the literature at various places; for a survey, see [11]. Important for the construction of Hadamard matrices are the symmetric C-matrices, of order v = 2 (mod 4), and the skew C-matrices, of order v = 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C-matrices. A more general class of matrices with zero diagonal is investigated, which contains the C-matrices and the matrices of (v, k, X)-systems on k and k + 1 in the… Expand
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References

SHOWING 1-10 OF 19 REFERENCES
Note on Hadamard's determinant theorem
Introduction. We shall call a square matrix A of order n an Hadamard matrix or for brevity an iî-matrix, if each element of A has the value ± 1 and if the determinant of A has the maximum possibleExpand
Strongly Regular Graphs of L2-type and of Triangular Type
This chapter discusses strongly regular graphs of L 2 -type and of triangular type. The theory of strongly regular graphs includes the theory of orthogonal latin squares. In a number of cases, it isExpand
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
Strongly regular graphs with (-1, 1, 0) adjacency matrix having eigenvalue 3
The ordinary graphs of finite order ν with (−1, 1, 0) adjacency matrix A satisfying ( A − ρ 1 I ) ( A − ρ 2 I ) = (ν − 1 + ρ 1 ρ 2 ) J , ρ 1 ρ 2 , AJ= ρ 0 J . The only eigenvalues are ρ 1 , ρ 2 , ρ 0Expand
Relations Among Generalized Hadamard Matrices, Relative Difference Sets, and Maximal Length Linear Recurring Sequences
It was established in (5) that the existence of a Hadamard matrix of order 4 t is equivalent to the existence of a symmetrical balanced incomplete block design with parameters v = 4 t — 1, k = 2 t —Expand
Finite permutation groups of rank 3
By the rank of a transitive permutation group we mean the number of orbits of the stabilizer of a point thus rank 2 means multiple transitivity. Interest is drawn to the simply transitive groups ofExpand
Strongly regular graphs, partial geometries and partially balanced designs.
This paper introduced the concept of a partial geometry, which serves to unify and generalize certain theorems on embedding of nets, and uniqueness of association schemes of partially balancedExpand
A Note on the Parameters of PBIB Association Schemes
This paper presents several results on association scheme parameters of partially balanced designs, dealing mostly with divisibility properties and based on the standard identities relating theseExpand
Some Aspects of Weighing Designs
Relative difference sets
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