On a generalization of spikes

We consider matroids with the property that every subset of the ground set of size t is contained in both an ℓ-element circuit and an ℓ-element cocircuit; we say that such a matroid has the (t, ℓ)-property. We show that for any positive integer t, there is a finite number of matroids with the (t, ℓ)-property for ℓ < 2t; however, matroids with the (t, 2t)-property form an infinite family. We say a matroid is a t-spike if there is a partition of the ground set into pairs such that the union of any t pairs is a circuit and a cocircuit. Our main result is that if a sufficiently large matroid has the (t, 2t)-property, then it is a t-spike. Finally, we present some properties of t-spikes.