Designs, Codes and Cryptography 29 (2003), 99-104 (proceedings of Finite Geometries, Oberwolfach 2001).
We show that 40 is the maximum number of points of a cap in AG(4,4). Up to semi-linear transformations there is only one such 40-cap. Its group of automorphisms is a semidirect product of an elementary abelian group of order 16 and the alternating group A_{5}.
The starting caps of the computer searches mentioned in the paper can be found here and an explicit example of the affine 40 cap here. The computer searches are analog to those used in 41 is the largest size of a cap in PG(4,4).
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