How Many Cards Should You Lay Out in Quad-128: A Classification of Caps in AG(7,2)

TL;DR

This paper classifies caps in the affine geometry AG(7,2), identifying the number and types of caps of sizes 10 to 12, and determining their completeness and maximum size.

Contribution

It provides a complete classification of caps in AG(7,2) for sizes ≥10, including their affine equivalence classes and completeness status.

Findings

Two classes of 10-caps identified

One class of 11-caps identified

One class of 12-caps that are complete and maximum size

Abstract

We define a cap in the affine geometry AG(n,2) to be a subset in which every collection of four points is in general position. In this paper, we classify, up to affine equivalence, all caps in AG(7,2) of size k greater than or equal to 10. In particular, we show that there are two equivalence classes of 10-caps and one equivalence class of 11-caps, none of which are complete, and one equivalence class of 12-caps, which are both complete and of maximum size.