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The graph of minimal distances of bent functions and its properties. (English) Zbl 1417.94138
Summary: A notion of the graph of minimal distances of bent functions is introduced. It is an undirected graph \((V,E)\) where \(V\) is the set of all bent functions in \(2k\) variables and \((f, g) \in E\) if the Hamming distance between \(f\) and \(g\) is equal to \(2^k\). It is shown that the maximum degree of the graph is equal to \(2^k (2^1 + 1) (2^2 + 1) \cdots (2^k + 1)\) and all its vertices of maximum degree are quadratic bent functions. It is obtained that the degree of a vertex from Maiorana-McFarland class is not less than \(2^{2k + 1} - 2^k\). It is proven that the graph is connected for \(2k = 2, 4, 6\), disconnected for \(2k \geq 10\) and its subgraph induced by all functions EA-equivalent to Maiorana-McFarland bent functions is connected.

94D10 Boolean functions
06E30 Boolean functions
Full Text: DOI
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