A coding theoretic approach to the uniqueness conjecture for projective planes of prime order

Abstract

An outstanding folklore conjecture asserts that, for any prime p, up to isomorphism the projective plane $PG(2,\mathbb {F}_p)$ over the field $\mathbb {F}_p := \mathbb {Z}/p\mathbb {Z}$ is the unique projective plane of order p. Let $\pi$ be any projective plane of order p. For any partial linear space ${\mathcal {X}}$, define the inclusion number $\mathbf{i}({\mathcal {X}},\pi )$ to be the number of isomorphic copies of ${\mathcal {X}}$ in $\pi$. In this paper we prove that if ${\mathcal {X}}$ has at most $\log _2 p$ lines, then $\mathbf{i}({\mathcal {X}},\pi )$ can be written as an explicit rational linear combination (depending only on ${\mathcal {X}}$ and p) of the coefficients of the complete weight enumerator (c.w.e.) of the p-ary code of $\pi$. Thus, the c.w.e. of this code carries an enormous amount of structural information about $\pi$. In consequence, it is shown that if $p > 2^ 9=512$, and $\pi$ has the same c.w.e. as $PG(2,\mathbb {F}_p)$, then $\pi$ must be isomorphic to $PG(2,\mathbb {F}_p)$. Thus, the uniqueness conjecture can be approached via a thorough study of the possible c.w.e. of the codes of putative projective planes of prime order.