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

Karianne Calta, Timothy E. Goldberg, Lauren L. Rose

TL;DR

This work classifies quad-free sets (caps) in the binary affine space $\operatorname{AG}(7,2)$, equivalently $\mathbb{Z}_2^7$, for sizes $k\ge10$ up to affine equivalence. It introduces extended-type and dependent-template frameworks to capture affine-dependence relations among cap elements, enabling explicit constructions and classifications. The authors establish two affine classes of $10$-caps, a single class of $11$-caps, and a single class of $12$-caps, with $11$-caps non-complete and $12$-caps complete and of maximum size $12$; no caps of size $13$ exist. The results link geometric coding theory in the binary setting to the cap-set/quad-free problem, yielding a complete threshold: to guarantee a quad in Quad-128 you must lay out at least $13$ cards.

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.

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

TL;DR

This work classifies quad-free sets (caps) in the binary affine space , equivalently , for sizes up to affine equivalence. It introduces extended-type and dependent-template frameworks to capture affine-dependence relations among cap elements, enabling explicit constructions and classifications. The authors establish two affine classes of -caps, a single class of -caps, and a single class of -caps, with -caps non-complete and -caps complete and of maximum size ; no caps of size exist. The results link geometric coding theory in the binary setting to the cap-set/quad-free problem, yielding a complete threshold: to guarantee a quad in Quad-128 you must lay out at least cards.

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.
Paper Structure (8 sections, 37 theorems, 78 equations, 2 figures, 3 tables)

This paper contains 8 sections, 37 theorems, 78 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Fundamental notions in $\mathop{\mathrm{AG}}\nolimits(n,2)$
  3. Quads and caps in $\mathop{\mathrm{AG}}\nolimits(n,2)$
  4. Caps of dimension $7$
  5. Caps of size $10$
  6. Caps of size 11
  7. Caps of size 12
  8. Caps of size 13

Key Result

Theorem 1.1

lamatpaper Four distinct cards $A, B, C, D$ form a quad if and only if as vectors, $A + B + C + D = \mathbf{0}$.

Figures (2)

  • Figure 1: Quad-64 Deck
  • Figure 2: A Quad-128 deck

Theorems & Definitions (83)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • ...and 73 more