Low-rank matrices with three prescribed off-diagonal values over $\mathbb{F}_p$

Let $p$ be a prime such that $2$ is a primitive root modulo $p$. We prove that, for every three-element set $L\subseteq\mathbb{F}_p$ and every integer $n\geq1$, there exists a symmetric $n\times n$ matrix over $\mathbb{F}_p$ with zero diagonal, off-diagonal entries in $L$, and rank $O_{p,L}(n^{1/3})$. This answers a question of Bukh for all such primes and, in particular, settles the case $L=\{1,4,6\}\subseteq\mathbb{F}_{19}$ singled out on his problem list. When $0\notin L$, a standard polynomial argument shows that the exponent $1/3$ is optimal. The proof combines a geometric construction in characteristic two with an incidence matrix factorization over $\mathbb{F}_p$. The same construction yields integer lifts $L'$ of the prescribed sets $L$ and symmetric $L'$-matrices of rational rank $O_{p,L}(n^{1/3})$. When $0\notin L'$, this improves Bukh's general quadratic lower bound, giving matrices of order $Ω_{p,L}(r^3)$ and rational rank at most $r$.