Demicaps in AG(4,3) and Their Relation to Maximal Cap Partitions
Abstract
In this paper, we introduce a fundamental substructure of maximal caps in the affine geometry AG(4,3) that we call demicaps. Demicaps provide a direct link to particular partitions of AG(4,3) into 4 maximal caps plus a single point. The full collection of 36 maximal caps that are in exactly one partition with a given cap C can be expressed as unions of two disjoint demicaps taken from a set of 12 demicaps; these 12 can also be found using demicaps in C. The action of the affine group on these 36 maximal caps includes actions related to the outer automorphisms of S6.
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