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


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Article Info

DATE

2019-03-07

AUTHORS

Bhaskar Bagchi

ABSTRACT

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 log2p 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>29=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. More... »

PAGES

1-15

References to SciGraph publications

  • 2008-01. Small weight codewords in the codes arising from Desarguesian projective planes in DESIGNS, CODES AND CRYPTOGRAPHY
  • 2012. On Characterizing Designs by Their Codes in BUILDINGS, FINITE GEOMETRIES AND GROUPS
  • 1954-11. On some finite non-desarguesian planes in ARCHIV DER MATHEMATIK
  • Identifiers

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    http://scigraph.springernature.com/pub.10.1007/s10623-019-00623-y

    DOI

    http://dx.doi.org/10.1007/s10623-019-00623-y

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1112609306


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