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Mathematics of Computation

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Orthomorphisms and the construction of projective planes

Authors: Felix Lazebnik and Andrew Thomason
Journal: Math. Comp. 73 (2004), 1547-1557
MSC (2000): Primary 05B15, 05C50, 05C62, 51E15, 68R10
Published electronically: July 31, 2003
MathSciNet review: 2047100
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Abstract: We discuss a simple computational method for the construction of finite projective planes. The planes so constructed all possess a special group of automorphisms which we call the group of translations, but they are not always translation planes. Of the four planes of order 9, three admit the additive group of the field $GF(9)$ as a group of translations, and the present construction yields all three. The known planes of order 16 comprise four self-dual planes and eighteen other planes (nine dual pairs); of these, the method gives three of the four self-dual planes and six of the nine dual pairs, including the ``sporadic'' (not translation) plane of Mathon.

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Additional Information

Felix Lazebnik
Affiliation: Department of Mathematical Sciences, Ewing Building, University of Delaware, Newark, Delaware 19716

Andrew Thomason
Affiliation: Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, United Kingdom

Keywords: Orthomorphisms, projective planes, translation
Received by editor(s): March 13, 2002
Received by editor(s) in revised form: January 22, 2003
Published electronically: July 31, 2003
Additional Notes: This research was supported partially by a grant from the London Mathematical Society.
Article copyright: © Copyright 2003 American Mathematical Society