Subgroups of Finite Fields As Cap Sets

TL;DR

The paper demonstrates that specific subgroups of finite fields, related to powers of elements, form maximal cap sets, with implications for combinatorial structures and related card games.

Contribution

It identifies new classes of maximal cap sets in finite fields derived from subgroup structures of powers, extending known results.

Findings

Subgroup of 20 nonzero fourth powers in GF(81) is a cap set.

Subgroup of 9 nonzero seventh powers in GF(64) is a cap set.

Certain multiplicative subgroups in fields of orders 243 and 729 are cap sets.

Abstract

We show the subgroup of 20 nonzero fourth powers in the finite field of order 81 is a cap set. Similarly, the subgroup of 9 nonzero seventh powers in the field of order 64 is a cap set. These are the cases related to the card games of SET and EvenQuads, and both are known to be maximal cap sets. A corollary is that the cosets of these subgroups form a partition by maximal caps of the multiplicative groups of their respective fields. We identify certain multiplicative subgroups of fields of orders 243 and 729 as cap sets, and show in general that the subgroup of $(2^n-1)$-th powers is a cap set in the field of order $2^{2n}$.