Abstract
In this paper, we explore the existence and nonlinearity of affine [Formula: see text]-cycle permutations on [Formula: see text] for different values of [Formula: see text] and [Formula: see text]. It is proved that a transposition [Formula: see text] on [Formula: see text] has nonlinearity [Formula: see text] for [Formula: see text], a [Formula: see text]-cycle permutation on [Formula: see text] has nonlinearity [Formula: see text] for [Formula: see text], [Formula: see text] a prime. The number of affine [Formula: see text]-cycle permutations on [Formula: see text] and a bound for their nonlinearity have been obtained. Some results on the cyclic decomposition of a permutation on [Formula: see text], [Formula: see text] a prime, have also been proved.
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