an:03981292
Zbl 0607.12009
Wan, Daqing
On a problem of Niederreiter and Robinson about finite fields
EN
J. Aust. Math. Soc., Ser. A 41, 336-338 (1986).
00152312
1986
j
11T06
polynomial over finite field; permutation polynomial; complete mapping polynomial; reduced degree
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 \textit{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.
H.Niederreiter
Zbl 0495.12018