In this paper, we study binomials having the form \(x^r(a+x^{3(q-1)})\) over the finite field \(\mathbb {F}_{q^2}\) with \(q=2^m\), and determine all the r’s and coefficients a’s making them permutations. For even m or odd m with \(3\not \mid m\), we prove that the characterization is necessary and sufficient. For the case of odd m and \(3\mid m\), we prove that the corresponding sufficient condition is also necessary for almost all r’s. Finally we obtain that the proportion of r’s we cannot prove the necessity is only about \(\frac{1}{40}\).

中文翻译：

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具有偶特征的有限域上的二项式置换

在本文中，我们研究了有限域\(\mathbb {F}_{q^2}\) 上形式为\(x^r(a+x^{3(q-1)})\) 的二项式用\(q=2^m\)，并确定所有r和系数a使它们排列。对于偶数m或奇数m与\(3\not \mid m\)，我们证明了表征是必要的和充分的。对于奇数m和\(3\mid m\) 的情况，我们证明了对应的充分条件对于几乎所有的r也是必要的。最后我们得到无法证明必要性的r的比例仅为\(\frac{1}{40}\)。