Partitions of AG(4,3) into Maximal Caps

TL;DR

This paper investigates how the 81 points of affine geometry AG(4,3) can be partitioned into maximal caps of 20 points each, revealing two classes of such partitions distinguished by their pairing structures under affine transformations.

Contribution

It characterizes the partitions of AG(4,3) into maximal caps, identifies their structure, and classifies them into two equivalence classes based on pairing properties.

Findings

Partition of AG(4,3) into 4 maximal caps plus a point P.

Two distinct classes of partitions based on pairing structures.

Structural properties of caps under affine transformations.

Abstract

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.