Hadamard Matrices related to Projective Planes

Unbalanced Hadamard Matrices and Finite Projective Planes of Even Order

By an unbalanced Hadamard matrix we mean a matrix H,, = (hii) such that (i) hii = l/fi or -I/fi, (ii) H, is orthogonal, and (iii) the number of positive entries exceeds the number of negative entries in each row. In particular it is well-known that the dimension n must be an even perfect square if the number of positive entries is the same in each row. It is easy to show that the number of posi...

Decompositions of Complete Multigraphs Related to Hadamard Matrices

Let bp(+Kv) be the minimum number of complete bipartite subgraphs needed to partition the edge set of +Kv , the complete multigraph with + edges between each pair of its v vertices. Many papers have examined bp(+Kv) for v 2+. For each + and v with v 2+, it is shown here that if certain Hadamard and conference matrices exist, then bp(+Kv) must be one of two numbers. Also, generalizations to deco...

Jacket matrices constructed from Hadamard matrices and generalized Hadamard matrices

A Path to Hadamard Matrices

There are characteristics of Hadamard matrices that enable an exhaustive search using algorithmic techniques. The search derives primarily from the eigenvalues which are constant after the Hadamard matrix is multiplied by its transpose. Generally this would be a performance concern but there are additional properties that enable the eigenvalues to be predicted. Here an algorithm is given to obt...

Smooth Projective Planes

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to CP. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high dimensional regular planes are lines, prove that homeomorphisms preserving plane curves are smooth collineations, and prove a variety of results analogous to the theory ...