A Class of Projective Planes of Cubic Order

A Class of Projective Planes

On cubic curves in projective planes of characteristic two

The aim of this paper is to examine various interesting results from the theory of general cub~c curves in projective planes of characteristic two. This leads to calculations involving nets of conics in the plane, invariants of the curves, syzygies, and Hessians. It is emphasized that classical methods, (that is those developed for geometries over fields of zero characteristic), do not always s...

Groups of generalised projectivities in projective planes of odd order

The generalised projectivities (GP's) associated with projective planes of odd order are investigated. These are non-singular linear mappings over GF(2) defined from the binary codes of these planes. One case that is investigated in detail corresponds to the group of an affine plane-every point corresponds to a GP. It is shown how many collineations that fix the line at infinity point-wise can ...

Graphs for Orthogonal Arrays and Projective Planes of Even Order

We consider orthogonal arrays of strength two and even order q having n columns which are equivalent to n − 2 mutually orthogonal Latin squares of order q. We show that such structures induce graphs on n vertices, invariant up to complementation. Previous methods worked only for single Latin squares of even order and were harder to apply. If q is divisible by four the invariant graph is simple ...

On projective planes of order less than 32

By our count, 245 projective planes of order less than 32 are currently known. This list is dominated by the 193 known planes of order 25. Most of these are either translation planes or Hughes planes, or planes obtained from these by the well-known process of repeatedly dualizing and deriving. We describe two new planes obtainable by the quite different method of ‘lifting quotients’. 2000 Mathe...