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Permutation polynomials and applications to coding theory. (English) Zbl 1107.11048
Using the characterization of permutation polynomials over the finite field \(F_q\) of the form \(X^rf(X^{(q-1)/d)}\) given by D. Wan and R. Lidl [Monatsh. Math. 112, No. 2, 149–163 (1991; Zbl 0737.11040)] the author exhibits a new class of permutation binomials. Moreover, he estimates the number of permutation binomials of the form \(X^r(X^{(q-1)/m}+a)\), \(a\in F_q\), using the Weil bound.
Finally, he gives some applications to coding theory mainly related to a conjecture of T. Helleseth [Discrete Math. 16, 209–232 (1976; Zbl 0348.94017)] on the existence of balanced words of the form \(Tr(x^k+ax)\), \(a\in F_{2^n}^*\), if \(\gcd(k,2^n-1)=1\).

MSC:
11T06 Polynomials over finite fields
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