In a geometry, a maximal cap is a collection of points of largest size containing no lines. In AG(4,3), maximal caps contain 20 points. The 81 points of AG(4,3) can be partitioned into 4 mutually disjoint maximal caps together with a single point P, where every pair of points that makes a line with P lies entirely inside one of those caps. The caps in a partition can be paired up so that both pairs are either in exactly one partition or they are both in two different partitions. This difference determines the two equivalence classes of partitions of AG(4,3) under the action by affine transformations.
@article{arxiv.1302.4703,
title = {Partitions of AG(4,3) into Maximal Caps},
author = {Michael Follett and Kyle Kalail and Elizabeth McMahon and Catherine Pelland and Robert Won},
journal= {arXiv preprint arXiv:1302.4703},
year = {2018}
}