New Nikodym set constructions over finite fields

For any fixed dimension $d\ge 3$ we construct a Nikodym set in $F_q^d$ of cardinality $q^d-(\frac{d-2}{\log 2}+1+o(1))q^{d-1}\log q$ in the limit $q\to \mathrm{\infty }$, when $q$ is an odd prime power. This improves upon the naive random construction, which gives a set of cardinality $q^d-(d-1+o(1))q^{d-1}\log q$, and is new in the regime where $F_q$ has unbounded characteristic and $q$ not a perfect square. While the final proofs are completely human generated, the initial ideas of the construction were inspired by output from the tools $\mathtt{\text{AlphaEvolve}}$ and $\mathtt{\text{DeepThink}}$. We also present a simple construction of Nikodym sets in $F_q^2$ for $q$ a perfect square that is a special case of known unital-based constructions, and matches the existing bounds of $q^2-q^{3\mathrm{/}2}+O(q\log q)$, assuming that $q$ is not the square of a prime $p\equiv 3(\mathrm{m}\mathrm{o}\mathrm{d}4)$.