The largest cap in \(AG\)(4, 4) and its uniqueness (Q1404318)

A \(k\)-cap in \(AG(n,q)\) or \(PG(n,q)\) is a set of \(k\) points no three of which are collinear. A \(k\)-cap is called complete when it is not contained in a larger cap. This article classifies the largest \(k\)-caps in \(AG(4,4)\). It is proven that the maximal size for a cap in \(AG(4,4)\) is equal to 40. When considered as an affine 40-cap in the projective space \(PG(4,4)\), there is only one orbit of affine 40-caps in \(PG(4,4)\) under the action of \(P\Gamma L(5,4)\) and two under the action of \(PGL(5,4)\). The cap is complete in \(PG(4,4)\). Its group of automorphisms has order 960, is a semidirect product of an elementary abelian group of order 16 and the alternating group \(A_5\), and acts transitively on the points of the 40-cap.