The p-Rank of the Incidence Matrix of Intersecting Linear Subspaces

  • Authors:

  • Peter Sin

  • Affiliations:

  • Department of Mathematics, University of Florida, Gainesville, FL 32611, U.S.A. sin@math.ufl.edu

  • Venue:
  • Designs, Codes and Cryptography
  • Year:
  • 2004

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Abstract

Let V be a vector space of dimension n+1 over a field of pt 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).