The smallest size of a complete cap in PG(3,7)PG(3,7)

We show that 17 is the smallest size of a complete cap in the projective space PG(3,7)PG(3,7) and that there are exactly four projectively inequivalent such caps. Along the way it is shown that a linear code [15,4,11]7[15,4,11]7 does not exist.