Cap set $r(7)$

The size of the largest set of points in $\mathbb{F}_3^{7}$ with no three on a line.

Verified bounds

Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.

• Lower bound: $236$ Checked construction
  cap-set-7-lower-236, submitted by daniel-munro, 2026-11-05
  [construction from Y. Edel and J. Bierbrauer, Large caps in small spaces, Designs, Codes and Cryptography 23 (2001) 197-212.]
• Upper bound: None yet

Updates from the literature

• 1994-07 Lower bound: $236$
  A.R. Calderbank and P.C. Fishburn, Maximal three-independent subsets of {0, 1, 2}^n, Designs, Codes and Cryptography 4 (1994) 203-211.
  [via I. Karapetyan and K. Karapetyan, New method for constructing complete cap sets, arXiv:2601.16917.]
• 2017-07 Upper bound: $291$
  N.D. Versluis, On the cap set problem: upper bounds on maximal cardinalities of caps in dimensions seven to ten, Bachelor's thesis, Delft University of Technology (2017).
  [via I. Karapetyan and K. Karapetyan, New method for constructing complete cap sets, arXiv:2601.16917.]
• 2022-06-20 Upper bound: $288$
  H.(M.)R. Thackeray, The cap set problem: up to dimension 7, arXiv:2206.09804.