  We consider matroids with the property that every subset of the ground set of
size $t$ is contained in both an $\ell$-element circuit and an $\ell$-element
cocircuit; we say that such a matroid has the $(t,\ell)$-property. We show that
for any positive integer $t$, there is a finite number of matroids with the
$(t,\ell)$-property for $\ell<2t$; 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.
