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

  • Daqing Wan (a1), Gary L. Mullen (a2) and Peter Jau-Shyong Shiue (a3)
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 cFq, 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.

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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

3. S. D. Cohen , Proof of a conjecture of Chowla and Zassenhaus on permutation polynomials, Canad. Math. Bull. 33 (1990), 230234.

7. A. B. Evans , Maximal sets of mutually orthogonal Latin squares, II, European J. Combin. 13 (1992), 345350.

11. M. D. Fried , R. Guralnick and J. Saxl , Schur covers and Carlitz's conjecture, Israel J. Math. 82 (1993), 157225.

12. D. Hachenberger and D. Jungnickel , Bruck nets with a transitive direction, Geom. Dedicata 36 (1990), 287313.

22. D. Wan and R. Lidl , Permutation polynomials of the form and their group structure, Monatsh. Math. 112 (1991), pp. 149163.

23. D. Wan , P. J.-S. Shiue and C. S. Chen , Value sets of polynomials over finite fields, Proc. Amer. Math. Soc. 119 (1993), 711717.

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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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