68,154 research outputs found
Intersection numbers for subspace designs
Intersection numbers for subspace designs are introduced and -analogs of
the Mendelsohn and K\"ohler equations are given. As an application, we are able
to determine the intersection structure of a putative -analog of the Fano
plane for any prime power . It is shown that its existence implies the
existence of a - subspace design. Furthermore, several
simplified or alternative proofs concerning intersection numbers of ordinary
block designs are discussed
Constructions of new matroids and designs over GF(q)
A perfect matroid design (PMD) is a matroid whose flats of the same rank all
have the same size. In this paper we introduce the q-analogue of a PMD and its
properties. In order to do that, we first establish a new cryptomorphic
definition for q-matroids. We show that q-Steiner systems are examples of
q-PMD's and we use this q-matroid structure to construct subspace designs from
q-Steiner systems. We apply this construction to S(2, 3, 13; q) q-Steiner
systems and hence establish the existence of subspace designs with previously
unknown parameters
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