Title: The $p$-rank of the Incidence Matrix of Intersecting Linear
Subspaces.


Authors:  Peter Sin
                        Department of Mathematics
                        University of Florida 
                        358 Little Hall
                        PO Box 118105
                        Gainesville, FL 32611-8105 



Abstract.

Let $V$ be a vector space of dimension $n+1$ over a field of $p^t$
elements.  A $d$-dimensional subspace and an $e$-dimensional subspace
are considered to be incident if their intersection is not the zero
subspace.  The rank of these incidence matrices, modulo $p$, are
computed for all $n$, $d$, $e$ and $t$. This result generalizes the
well known formula of Hamada for the incidence matrices between points
and subspaces of given dimensions in a finite projective space. A
generating function for these ranks as $t$ varies, keeping $n$, $d$
and $e$ fixed, is also given.  In the special case where the
dimensions are complementary, i.e. $d+e=n+1$, our formula improves
previous upper bounds on the size of partial $m$-systems (as defined
by Shult and Thas).

Preprint Nov. 14, 2002