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

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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).

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Authors and Affiliations

  1. Department of Mathematics, University of Florida, Gainesville, FL, 32611, U.S.A.

    Peter Sin

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  1. Peter Sin
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Sin, P. The p-Rank of the Incidence Matrix of Intersecting Linear Subspaces. Designs, Codes and Cryptography 31, 213–220 (2004). https://doi.org/10.1023/B:DESI.0000015884.34185.4a

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  • DOI: https://doi.org/10.1023/B:DESI.0000015884.34185.4a

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