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.