The 20 nonzero fourth powers of GF(81) form a maximal SET cap in AG(4,3) — no three distinct points sum to 0 (mod 3) — and the 9 nonzero seventh powers of GF(64) form a maximal EvenQuads cap in AG(6,2) — no four distinct points sum to 0 (mod 2) (Kable–Mills–Wright: subgroups of finite fields as cap sets)
three distinct indices i<j<k with set81[i]+set81[j]+set81[k] ≡ 0 (mod 3), or four distinct indices with the corresponding eq64 sum ≡ 0 (mod 2)