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Distance duality on some classes of Boolean functions. (English) Zbl 1432.94228
Summary: A function $$f$$ is at a distance $$d(f, \mathcal S)$$ from a set $$\mathcal S$$ of Boolean functions if the minimum Hamming distance between $$f$$ and all $$g\in\mathcal S$$ is $$d(f, \mathcal S)$$. Given a set $$\mathcal S$$ of Boolean functions, the set $$\hat{\mathcal S}$$ of functions is said to have a maximum distance from $$\mathcal S$$ if it has the property that for all $$f\in \hat{\mathcal S}$$, $$d(f, \mathcal S)$$ is maximum. The well-studied bent Boolean functions $$(\mathcal S)$$ are defined to have a maximum distance from the set of affine functions $$(\hat{ \mathcal S})$$. N. Tokareva [Bent functions. Results and applications to cryptography. Amsterdam: Elsevier/Academic Press (2015; Zbl 1372.94002)] showed the converse is also true. That is, $$\mathcal S$$ is a maximum distance from $$\hat{\mathcal S}$$. In such a case, we say that $$\mathcal S$$ and $$\hat{\mathcal S}$$ are mutually maximally distant.
We introduce partition set functions, which include symmetric functions, rotation symmetric functions, self-anti-dual-functions, linear structure functions, and degenerate functions. We show that partition set functions associated with a partition on the domain $$\mathbb F^n_2$$ are mutually maximally distant from another set $$\hat{\mathcal S}$$ of functions. We show that the distributions of weights in $$\hat{\mathcal S}$$ is binomial and centered at $$2^{n-1}$$, the point at which a function is perfectly balanced.
Because there is much interest in symmetric functions, we consider this special case, and verify a prior enumeration of functions that are maximally distant from symmetric functions [I. Ivchenko, Yu. I. Medvedev, and V. A. Mironova, “Symmetric Boolean functions and their metric properties matrices of transitions of differences when using some modular groups” (Russian), Mat. Vopr. Kriptogr. 4, No. 4, 49–63 (2013)] (we call them maximally asymmetric functions). We characterize balanced maximally asymmetric functions. A similar analysis is done on rotation symmetric functions.

##### MSC:
 94D10 Boolean functions 06E30 Boolean functions