### Abstract

A function/is at a distance d(f,S) from a set S of Boolean functions if the minimum Hamming distance between/and all g ? S is d(f,S). Given a set S of Boolean functions, the set S of functions is said to have a maximum distance from S if it has the property that for all/? S, d(f,S) is maximum. The well-studied bent Boolean functions (S) are defined to have a maximum distance from the set of affine functions (5). Tokareva [14] showed the converse is also true. That is, 5 is a maximum distance from S. In such a case, we say that <S and <S are mutually maximally distant. We introduce partition set functions, which include symmetric functions, rotation symmetric functions, self-anti-dual-functions, lin-ear structure functions, and degenerate functions. We show that partition set functions associated with a partition on the domain FJ are mutually maximally distant from another set S of functions. We show that the distributions of weights in 5 is binomial and centered at 2n 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 [8] (we call them maximally asymmetric functions). We characterize balanced maximally asymmetric functions. A similar analysis is done on rotation symmetric functions.

Original language | English |
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Pages (from-to) | 181-198 |

Number of pages | 18 |

Journal | Journal of Combinatorial Mathematics and Combinatorial Computing |

Volume | 107 |

Publication status | Published - Nov 2018 |

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## Cite this

*Journal of Combinatorial Mathematics and Combinatorial Computing*,

*107*, 181-198.