A generalization of a covering problem of mullin and stanton for matroids

A (q,m,k) cover of V(k,q), the vector space of k-tuples over GF(q), is a subset of the non-zero vectors of V(k,q) which has rank k and has non-empty intersection with every subspace of V(k,q) of rank k−m. A (q,m,k) cover may also be viewed as a matroid. As such it is essentially the image in V(k,q) of a restriction M of PG(k−1,q) under some representation, where M has rank k and critical exponent greater than m. An earlier paper answered several questions of Mullin and Stanton concerning (2,m,k) covers. This paper answers the corresponding questions for (q,m,k) covers when q>2. In particular, the least number η(q,m,k) of elements in a (q,m,k) cover is determined and those matroids having exactly η(q,m,k) elements are characterized.