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The number of permutation polynomials of the form f(x) cx over a finite field

Published online by Cambridge University Press:  20 January 2009

Daqing Wan ,
Gary L. Mullen  and
Peter Jau-Shyong Shiue
Daqing Wan
Affiliation:
Department of Mathematical SciencesUniversity of Nevada, Las Vegas, NV 89154USA e-mail: [email protected]
Gary L. Mullen
Affiliation:
Department of MathematicsThe Pennsylvania State UniversityUniversity Park, PA 16802USA e-mail: [email protected]
Peter Jau-Shyong Shiue
Affiliation:
Department of Mathematical SciencesUniversity of Nevada. Las Vegas, NV 89154USA e-mail: [email protected]
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Abstract

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Let Fq be the finite field of q elements. Let f(x) be a polynomial of degree d over Fq and let r be the least non-negative residue of q-1 modulo d. Under a mild assumption, we show that there are at most r values of c∈Fq, such that f(x) + cx is a permutation polynomial over Fq. This indicates that the number of permutation polynomials of the form f(x) +cx depends on the residue class of q–1 modulo d.

As an application we apply our results to the construction of various maximal sets of mutually orthogonal latin squares. In particular for odd q = pn if τ(n) denotes the number of positive divisors of n, we show how to construct τ(n) nonisomorphic complete sets of orthogonal squares of order q, and hence τ(n) nonisomorphic projective planes of order q. We also provide a construction for translation planes of order q without the use of a right quasifield.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 1 , February 1995 , pp. 133 - 149
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

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