Solution to a problem of S. Payne
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- by Xiang-dong Hou PDF
- Proc. Amer. Math. Soc. 132 (2004), 1-6 Request permission
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
- James A. Davis and Qing Xiang, A family of partial difference sets with Denniston parameters in nonelementary abelian 2-groups, European J. Combin. 21 (2000), no. 8, 981–988. MR 1797679, DOI 10.1006/eujc.2000.0416
- P. Dembowski, Finite geometries, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 44, Springer-Verlag, Berlin-New York, 1968. MR 0233275
- Leonard Eugene Dickson, Linear groups: With an exposition of the Galois field theory, Dover Publications, Inc., New York, 1958. With an introduction by W. Magnus. MR 0104735
- Rudolf Lidl and Harald Niederreiter, Finite fields, Encyclopedia of Mathematics and its Applications, vol. 20, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn. MR 746963
- Stanley E. Payne, Affine representations of generalized quadrangles, J. Algebra 16 (1970), 473–485. MR 273503, DOI 10.1016/0021-8693(70)90001-3
- S. E. Payne, Linear transformations of a finite field, Amer. Math. Monthly 78 (1971), 659 – 660.
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
- Email: xhou@euler.math.wright.edu
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 1-6
- MSC (2000): Primary 11T06; Secondary 51E20
- DOI: https://doi.org/10.1090/S0002-9939-03-07240-X
- MathSciNet review: 2021242