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

Article Type

Research Article

Publication Title

Designs, Codes, and Cryptography


An outstanding folklore conjecture asserts that, for any prime p, up to isomorphism the projective plane PG(2 , Fp) over the field Fp: = Z/ pZ is the unique projective plane of order p. Let π be any projective plane of order p. For any partial linear space X, define the inclusion number i(X, π) to be the number of isomorphic copies of X in π. In this paper we prove that if X has at most log 2p lines, then i(X, π) can be written as an explicit rational linear combination (depending only on X and p) of the coefficients of the complete weight enumerator (c.w.e.) of the p-ary code of π. Thus, the c.w.e. of this code carries an enormous amount of structural information about π. In consequence, it is shown that if p> 2 9= 512 , and π has the same c.w.e. as PG(2 , Fp) , then π must be isomorphic to PG(2 , Fp). 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.

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Open Access, Green