In this paper, we first study biplanes D with parameters (v, k, 2), where the block size k ∈ {13, 16}. These are the smallest parameter values for which a classification is not available. We show that if k = 13, then either D is the Aschbacher biplane or its dual, or Aut(D) is a subgroup of the cyclic group of order 3. In the case where k = 16, we prove that |Aut(D)| divides 27 · 32 · 5 · 7 · 11 · 13. We also provide an example of a biplane with parameters (16, 6, 2) with a flag-transitive and point-primitive subgroup of automorphisms preserving a homogeneous cartesian decomposition. This motivated us to study biplanes with point- primitive automorphism groups preserving a cartesian decomposition. We prove that such an automorphism group is either of affine type (as in the example), or twisted wreath type.
