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A note on constructing permutation polynomials. (English) Zbl 1215.11112
Let $$H$$ be a subgroup of order $$d$$ in $$\mathbb F_q^*$$. Starting with a polynomial $$T$$ which maps $$\mathbb F_q$$ into $$H\cup\{ 0\}$$ and a polynomial $$f$$ which maps $$H$$ onto a coset $$Ha$$, which satisfy certain simple conditions, the authors show that $$F(x)=x^s(x^{-s}f)(T(x))$$ is a permutation polynomial. When $$q-1=nd$$ and $$n\equiv 1\pmod{d}$$ then $$T(x)=x^n$$ satisfies the requirements and their construction yields some of the permutations of D. Wan and R. Lidl [Monatsh. Math. 112, No. 2, 149–163 (1991; Zbl 0737.11040)]. When $$n\not\equiv 1\pmod{d}$$ then the authors show by example that their permutations are not of the Wan-Lidl form arise.
If $$\mathbb F_{q'}\subset \mathbb F_q$$ and $$H=\mathbb F_{q'}^*$$ then the construction is a theorem on lifting permutation polynomials from $$\mathbb F_{q'}$$ to $$\mathbb F_q$$. They also give a second lifting theorem which is similar.

##### MSC:
 11T06 Polynomials over finite fields
##### Keywords:
permutation polynomial; finite field
Full Text:
##### References:
 [1] A. Blokhuis, R.S. Coulter, M. Henderson, C.M. O’Keefe, Permutations amongst the Dembowski-Ostrom polynomials, in: D. Jungnickel, H. Niederreiter (Eds.), Finite Fields and Applications: Proceedings of the Fifth International Conference on Finite Fields and Applications, 2001, pp. 37-42 · Zbl 1009.11064 [2] Kantor, W.M., Spreads, translation planes and kerdock sets I and II, SIAM J. alg. discrete math., 3, 151-165, (1982), 308-318 · Zbl 0493.51008 [3] Lidl, R.; Niederreiter, H., Finite fields, Encyclopedia math. appl., vol. 20, (1983), Addison-Wesley Reading, now distributed by Cambridge University Press [4] Mullen, G.L., Permutation polynomials: A matrix analogue of Schur’s conjecture and a survey of recent results, Finite fields appl., 1, 242-258, (1995) · Zbl 0828.11070 [5] Wan, D.; Lidl, R., Permutation polynomials of the form $$x^r f(x^{(q - 1) / d})$$ and their group structure, Monatsh. math., 112, 149-163, (1991) · Zbl 0737.11040
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