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On a problem of Niederreiter and Robinson about finite fields. (English) Zbl 0607.12009
A polynomial f(x) over a finite field \({\mathbb{F}}_ q\) is called a permutation polynomial of \({\mathbb{F}}_ q\) if the mapping induced by f(x) is a permutation of \({\mathbb{F}}_ q\). If both f(x) and \(f(x)+x\) are permutation polynomials of \({\mathbb{F}}_ q\), then f(x) is called a complete mapping polynomial of \({\mathbb{F}}_ q\). The degree of the reduction of f(x) modulo \(x^ q-x\) is called the reduced degree of f(x).
The reviewer and K. H. Robinson [ibid. 33, 197-212 (1982; Zbl 0495.12018)] have shown that for a finite field \({\mathbb{F}}_ q\) with odd order \(q>3\), any complete mapping polynomial has reduced degree at most q-3. In the present paper this result is proved for finite fields \({\mathbb{F}}_ q\) of even order \(q>3\). The proof is based on a clever extension of the method for odd q.
Reviewer: H.Niederreiter

11T06 Polynomials over finite fields