Classification of maximal caps in PG(3,5) different from elliptic quadrics

Ak-cap in PG(3,q) is a set of k points, no three of which are collinear. A k-cap is calledcomplete if it is not contained in a (k+1)-cap. The maximum valuem ₂(3, q) ofk for which there exists a k-cap in PG(3,q) is q²+1. Letm ₂(3, q) denote the size of the second largest complete k-cap in PG(3,q). This number is only known for the smallest values of q, namely for q=2, 3,4 (cf. 2], pp. 96–97 and 3], p. 303). In this paper we show thatm ₂(3,5)=20. We also prove that there are, up to isomorphism, only two complete 20-caps in PG(3,5) and determine their collineation groups.In memoriam Giuseppe TalliniWork done within the activity of GNSAGA of CNR and supported by MURST.