A bent function is a Boolean function in even number of variables which is on the maximal Hamming distance from the set of affine Boolean functions. It is called self-dual if it coincides with its dual. It is called anti-self-dual if it is equal to the negation of its dual. A mapping of the set of all Boolean functions in n variables to itself is said to be isometric if it preserves the Hamming distance. In this paper we study isometric mappings which preserve self-duality and anti-self-duality of a Boolean bent function. The complete characterization of these mappings is obtained for \(n\geqslant 4\). Based on this result, the set of isometric mappings which preserve the Rayleigh quotient of the Sylvester Hadamard matrix, is characterized. The Rayleigh quotient measures the Hamming distance between bent function and its dual, so as a corollary, all isometric mappings which preserve bentness and the Hamming distance between bent function and its dual are described.

中文翻译：

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自对折函数集合的自同构群

弯曲函数是具有偶数个变量的布尔函数，该变量与仿射布尔函数集之间的最大汉明距离。如果它与对偶一致，则称为自对偶。如果它等于其对偶的否定，则称为反自我对偶。如果它保留汉明距离，则将n个变量中所有布尔函数的集合映射到自身是等轴测的。在本文中，我们研究了等距映射，该映射保留了布尔弯曲函数的自对偶性和反自对偶性。这些映射的完整特征是针对\（n \ geqslant 4 \）获得的。基于此结果，对保留Sylvester Hadamard矩阵的Rayleigh商的等距映射集进行了表征。瑞利商测量折弯函数与其对偶之间的汉明距离，因此，必然地介绍了所有保留折弯度的等距映射以及折弯函数与其对偶之间的汉明距离。