Suppose that $ \mu_p $ is a probability measure defined on the input space of Boolean functions. We consider a generalization of Walsh–Hadamard transform on Boolean functions to $ \mu_p $-Walsh–Hadamard transforms. In this paper, first, we derive the properties of $ \mu_p $-Walsh–Hadamard transformation for some classes of Boolean functions and specify a class of nonsingular affine transformations that preserve the $ \mu_p $-bent property. We further derive the results on $ \mu_p $-Walsh–Hadamard transform of concatenation of Boolean functions and provide some secondary constructions of $ \mu_p $-bent functions. Finally, we discuss the $ \mu_p $-bentness for Maiorana–McFarland class of bent functions.

中文翻译：

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关于弯曲布尔函数泛化的注释

假设$ \ mu_p $是在布尔函数的输入空间上定义的概率测度。我们考虑将布尔函数上的Walsh–Hadamard变换推广为$ \ mu_p $ -Walsh–Hadamard变换。在本文中，首先，我们为某些布尔函数类推导$ \ mu_p $ -Walsh–Hadamard变换的属性，并指定一类保留$ \ mu_p $ -bent属性的非奇异仿射变换。我们进一步推导布尔函数级联的$ \ mu_p $ -Walsh–Hadamard变换的结果，并提供$ \ mu_p $ -bent函数的一些辅助构造。最后，我们讨论弯曲函数的Maiorana–McFarland类的$ \ mu_p $-弯曲度。