Cap set $r(8)$

The size of the largest set of points in $\mathbb{F}_3^{8}$ 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: $512$ Checked construction
  cap-set-8-lower-512, submitted by daniel-munro, 2026-11-05
  [construction from B. Romera-Paredes, M. Barekatain, A. Novikov, M. Balog, M.P. Kumar, E. Dupont, F.J.R. Ruiz, J.S. Ellenberg, P. Wang, O. Fawzi, P. Kohli and A. Fawzi, Mathematical discoveries from program search with large language models, Nature 625 (2024) 468-475.]
• Upper bound: None yet

Updates from the literature

• 2001-07 Lower bound: $496$
  Y. Edel and J. Bierbrauer, Large caps in small spaces, Designs, Codes and Cryptography 23 (2001) 197-212.
  [via B. Romera-Paredes, M. Barekatain, A. Novikov, M. Balog, M.P. Kumar, E. Dupont, F.J.R. Ruiz, J.S. Ellenberg, P. Wang, O. Fawzi, P. Kohli and A. Fawzi, Mathematical discoveries from program search with large language models, Nature 625 (2024) 468-475.]
• 2017-07 Upper bound: $771$
  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.]
• 2023-12-14 Lower bound: $512$
  B. Romera-Paredes, M. Barekatain, A. Novikov, M. Balog, M.P. Kumar, E. Dupont, F.J.R. Ruiz, J.S. Ellenberg, P. Wang, O. Fawzi, P. Kohli and A. Fawzi, Mathematical discoveries from program search with large language models, Nature 625 (2024) 468-475.
  [via I. Karapetyan and K. Karapetyan, New method for constructing complete cap sets, arXiv:2601.16917.]