On the distribution of orbits of PGL2(q) in Fqn and the Klapper conjecture

Igor E. Shparlinski

doi:10.1137/090770746

On the distribution of orbits of PGL2(q) in Fqn and the Klapper conjecture

Igor E. Shparlinski

School of Computing

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Abstract

Motivated by a conjecture of Klapper [Finite Fields, Coding Theory, and Advances in Communications and Computing, Marcel Dekker, New York, 1993], we study the distribution of elements ζ of a finite field F of qⁿ elements under the action of the transformations ? -· (aζ + b)/(cζ + d) for matrices ( acbd) ε PGL₂(q). We slightly improve a result of Niederreiter and Winterhof [Finite Fields Appl, 9 (2003), pp. 458-471] towards this conjecture. On the other hand, we also show that the original conjecture is false as stated.

Original language	English
Pages (from-to)	2093-2099
Number of pages	7
Journal	SIAM Journal on Discrete Mathematics
Volume	23
Issue number	4
DOIs	https://doi.org/10.1137/090770746
Publication status	Published - 2009

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Copyright SIAM Publications. Article archived for private and non-commercial use with the permission of the author and according to publisher conditions. For further information see http://www.siam.org/

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abstract = "Motivated by a conjecture of Klapper [Finite Fields, Coding Theory, and Advances in Communications and Computing, Marcel Dekker, New York, 1993], we study the distribution of elements ζ of a finite field F of qn elements under the action of the transformations ? -· (aζ + b)/(cζ + d) for matrices ( acbd) ε PGL2(q). We slightly improve a result of Niederreiter and Winterhof [Finite Fields Appl, 9 (2003), pp. 458-471] towards this conjecture. On the other hand, we also show that the original conjecture is false as stated.",

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On the distribution of orbits of PGL2(q) in Fqn and the Klapper conjecture

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