Lower bounds for the cardinality of minimal blocking sets in projective spaces Original Research Article

In this paper new lower bounds for the cardinality of minimal m-blocking sets are determined. Let r2(q) be the number such that q+r2(q)+1 is the cardinality of the smallest non-trivial line-blocking set in a plane of order q. If B is a minimal m-blocking set in PG(n,q) that contains at most qm+qm−1+…+q+1+r2(q)·(∑i=2m−n′m−1qi) points for an integer n′ satisfying m⩽n′⩽2m, then the dimension of 〈B〉 is at most n′. If the dimension of 〈B〉 is n′, then the following holds. The cardinality of B equals qm+qm−1+…+q+1+r2(q)(∑i=2m−n′m−1qi). For n′=m the set B is an m-dimensional subspace and for n′=m+1 the set B is a cone with an (m−2)-dimensional vertex over a non-trivial line-blocking set of cardinality q+r2(q)+1 in a plane skew to the vertex. This result is due to Heim (Mitt. Math. Semin. Giessen 226 (1996), 4–82). For n′>m+1 and q not a prime the number q is a square and for q⩾16 the set B is a Baer cone. If q is odd and |B|