A Complete Characterization of Plateaued Boolean Functions in Terms of Their Cayley Graphs

Abstract

In this paper we find a complete characterization of plateaued Boolean functions in terms of the associated Cayley graphs. Precisely, we show that a Boolean function f is s-plateaued (of weight = 2 (n+s−2)/2 ) if and only if the associated Cayley graph is a complete bipartite graph between the support of f and its complement (hence the graph is strongly regular of parameters e = 0, d = 2 (n+s−2)/2 ). Moreover, a Boolean function f is s-plateaued (of weight = 2 (n+s−2)/2 ) if and only if the associated Cayley graph is strongly 3-walk-regular (and also strongly -walk-regular, for all odd ≥ 3) with some explicitly given parameters.

Three-weight codes, triple sum sets, and strongly walk regular graphs

“…The second set admits eight possible triples of frequencies which reduce to 4 after using 5 ≤ k ≤ 8. These triples are [16,12,3] These are B K LC(G F(2), 12, 5) and B K LC(G F(2), 12, 6) respectively. We do not know if they are unique with these weight distributions.…”

“…In [16] it is proved that so-called plateaued Boolean functions f on F n 2 have a Cayley graph G f on the group (F n 2 , +) with generating set the support of f that is…”