Proceedings of the Edinburgh Mathematical Society (Series 2)

Research Article

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

Daqing Wana1*, Gary L. Mullena2** and Peter Jau-Shyong Shiuea3*

a1 Department of Mathematical Sciences University of Nevada, Las Vegas, NV 89154 USA e-mail: dwan@nevada.edu

a2 Department of Mathematics The Pennsylvania State University University Park, PA 16802 USA e-mail: mullen@math.psu.edu

a3 Department of Mathematical Sciences University of Nevada. Las Vegas, NV 89154 USA e-mail: shiue@nevada.edu

Abstract

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 cxs2208Fq, 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.

(Received June 17 1993)

Footnotes

* These authors would like to thank UNLV for partial support by the University Research Grants and Fellowship Committee. The first author would also like to thank the Institute for Advanced Study for its hospitality, and the National Science Foundation and the UNLV Faculty Development Leave Committee for partial support.

** This author would like to thank the National Security Agency for partial support under grant agreement #MDA904-92-H-3044.