G. Eric Moorhouse

Some p -Ranks Related to Orthogonal Spaces

AbstractWe determine the p-rank of the incidence matrix of hyperplanes of PG(n, pe) and points of a nondegenerate quadric. This yields new bounds for ovoids and the size of caps in finite orthogonal spaces. In particular, we show the nonexistence of ovoids in

Some p-ranks related to Hermitian varieties

O_{10}^ + (2^e ),O_{10}^ + (3^e ),O_9 (5^e ),O_{12}^ + (5^e )

Two-Graphs and Skew Two-Graphs in Finite Geometries

and

Approaching Some Problems in Finite Geometry Through Algebraic Geometry

O_{12}^ + (7^e )

The 2-Transitive Complex Hadamard Matrices

. We also give slightly weaker bounds for more general finite classical polar spaces. Another application is the determination of certain explicit bases for the code of PG(2, p) using secants, or tangents and passants, of a nondegenerate conic.

Subplanes of order 3 in Hughes Planes

Numerous computational examples suggest that if ℛk-1 ⊂ ℛk are (k- 1)- and k-nets of order n, then rankp ℛk - rankp ℛk-1 ≥ n - k + 1 for any prime p dividing n at most once. We conjecture that this inequality always holds. Using characters of loops, we verify the conjecture in case k = 3, proving in fact that if pe ∥ n, then rankp ℛ3 ≥ 3n - 2 - e, where equality holds if and only if the loop G coordinatizing ℛ3 has a normal subloop K such that G/K is an elementary abelian group of order pe. Furthermore if n is squarefree, then rankp ℛ = 3n - 3 for every prime p ¦ n, if and only if ℛ3 is cyclic (i.e., ℛ3 is coordinated by a cyclic group of order n).The validity of our conjectured lower bound would imply that any projective plane of squarefree order, or of order n ≡ 2 mod 4, is in fact desarguesian of prime order.Finally, our conjectured lower bound holds with equality in the case of desarguesian nets (i.e., subnets of AG(2, p)), which leads to an easy description of an explicit basis for the Fp-code of AG(2, p).