The graph of minimal distances of bent functions and its properties

Abstract

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 \(2k \ge 10\) and its subgraph induced by all functions EA-equivalent to Maiorana—McFarland bent functions is connected.

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