I've been reading about cap sets in $\mathbb{F}_3^d $ over the past couple of days and wondered why we can only find bounds for sizes of cap sets for $d>6$. The proofs are somewhat unclear to me (as a physics student) so I couldn't extrapolate from there.

I've been reading about cap sets in $\mathbb{F}_3^d $ over the past couple of days and wondered why we can only find bounds, as opposed to exact values, for (maximum) sizes of cap sets for $d>6$. The proofs are somewhat unclear to me (as a physics student) so I couldn't extrapolate from there.

The largest sizes of cap sets in $\mathbb{F}_3^d$ for $d \leq 6$ are listed at https://oeis.org/A090245.