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    Gaeta, Mario Matteo, Olivia Di Klimov, Andrei B and Guise, Hubert de 2014. Discrete phase-space approach to mutually orthogonal Latin squares. Journal of Physics A: Mathematical and Theoretical, Vol. 47, Issue. 43, p. 435303.


    2013. Handbook of Finite Fields.


    Ball, Simeon and Gács, András 2009. On the graph of a function over a prime field whose small powers have bounded degree. European Journal of Combinatorics, Vol. 30, Issue. 7, p. 1575.


    Ball, Simeon Gács, András and Sziklai, Peter 2008. On the number of directions determined by a pair of functions over a prime field. Journal of Combinatorial Theory, Series A, Vol. 115, Issue. 3, p. 505.


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  • Proceedings of the Edinburgh Mathematical Society, Volume 38, Issue 1
  • February 1995, pp. 133-149

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)
  • DOI: http://dx.doi.org/10.1017/S001309150000626X
  • Published online: 01 January 2009
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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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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