The minimum-weight dual-code characterization of the Desarguesian plane

Let $\pi$ be a projective plane of prime order $p$, and let $C_p^\perp(\pi)$ denote the dual of its $p$-ary code. For a codeword, its Hamming weight is the number of nonzero coordinates. The minimum-weight dual-code conjecture. The code $C_p^\perp(\pi)$ has at most

$$p^2(p^3-1)\binom{p+1}{3}$$

words of Hamming weight $3p-3$, and equality holds if and only if $\pi$ is isomorphic to $PG(2,\mathbb{F}_p)$. The claim is proposed as a coding-theoretic route toward the uniqueness conjecture for projective planes of prime order; no resolution is given in the supplied text.