A Permutation Problem for Finite Fields

Alan R. Prince
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Alan R. Prince: Heriot-Watt University, Department of Mathematics

A chapter in Finite Fields and Applications, 2001, pp 409-417 from Springer

Abstract: Abstract We consider the block structure of the incidence matrix of a projective plane of order p + 1 which admits a collineation of order p with three fixed points, where p is a prime. We show that, if p ≡ 3 (mod 4), p 2 × p 2 block in the incidence matrix can always be completed. The construction utilises the square root map on the quadratic residues mod p. The problem leads to a much more general question about the existence of a certain type of permutation of the nonzero elements of a finite field GF(q). The existence of a permutation with the required properties would lead to a construction of a projective plane of order q + 1.

Keywords: Projective Plane; Finite Field; Incidence Matrix; Quadratic Residue; Circulant Matrix (search for similar items in EconPapers)
Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-56755-1_31

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DOI: 10.1007/978-3-642-56755-1_31

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