Non-existence of a simple 3-(16, 7, 5) design with an automorphism of order 3

Using recent results concerning tactical decompositions of tt-designs with t>2t>2, we make a step forward on the long-standing question about the existence of a simple 3-(16, 7, 5) design; if such a design exists, then its full automorphism group has order a power of 2, possibly equal to 1.