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.
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References
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© 2001 Springer-Verlag Berlin Heidelberg
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Prince, A.R. (2001). A Permutation Problem for Finite Fields. In: Jungnickel, D., Niederreiter, H. (eds) Finite Fields and Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56755-1_31
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DOI: https://doi.org/10.1007/978-3-642-56755-1_31
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-62498-8
Online ISBN: 978-3-642-56755-1
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