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Solution to a problem of S. Payne

Author: Xiang-dong Hou
Journal: Proc. Amer. Math. Soc. 132 (2004), 1-6
MSC (2000): Primary 11T06; Secondary 51E20
Published electronically: August 13, 2003
MathSciNet review: 2021242
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Abstract | References | Similar Articles | Additional Information

Abstract: A problem posed by S. Payne calls for determination of all linearized polynomials $f(x)\in\mathbb{F} _{2^n}[x]$ such that $f(x)$and $f(x)/x$ are permutations of $\mathbb{F} _{2^n}$ and $\mathbb{F} _{2^n}^*$respectively. We show that such polynomials are exactly of the form $f(x)=ax^{2^k}$ with $a\in\mathbb{F} _{2^n}^*$ and $(k,n)=1$. In fact, we solve a $q$-ary version of Payne's problem.

References [Enhancements On Off] (What's this?)

  • 1. J. Davis and Q. Xiang, A family of partial difference sets with Denniston parameters in nonelementary abelian 2-groups, European J. Combin. 21 (2000), 981 - 988. MR 2002a:05043
  • 2. P. Dembowski, Finite Geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Springer-Verlag, New York, 1968. MR 38:1597
  • 3. L. E. Dickson, Linear Groups: With an Exposition of the Galois Field Theory, Dover, New York, 1958. MR 21:3488
  • 4. R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, Reading, MA, 1983. MR 86c:11106
  • 5. S. E. Payne, Affine representations of generalized quadrangles, J. Algebra 16 (1970), 473 - 485. MR 42:8381
  • 6. S. E. Payne, Linear transformations of a finite field, Amer. Math. Monthly 78 (1971), 659 - 660.

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Additional Information

Xiang-dong Hou
Affiliation: Department of Mathematics and Statistics, Wright State University, Dayton, Ohio 45435
Address at time of publication: Department of Mathematics, University of South Florida, Tampa, Florida 33620

Keywords: Finite field, linearized polynomial, permutation polynomial
Received by editor(s): July 29, 2002
Published electronically: August 13, 2003
Additional Notes: This research was supported by NSA grant MDA 904-02-1-0080
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2003 American Mathematical Society

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