Block sensitivity of minterm-transitive functions

Boolean functions with symmetry properties are interesting from a complexity theory perspective; extensive research has shown that these functions, if nonconstant, must have high ‘complexity’ according to various measures. In recent work of this type, Sun gave bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy bs(f) = Ω(N1/3), and that there exists such a function for which bs(f) = O(N3/7 lnN). His example function belongs to a subclass of transitively invariant functions called the minterm-transitive functions (defined in earlier work by Chakraborty). We extend these results in two ways. First, we show that nonconstant mintermtransitive functions satisfy bs(f) = Ω(N3/7). Thus Sun’s example function has nearly minimal block sensitivity for this subclass. Second, we give an improved example: a minterm-transitive function for which bs(f) = O(N3/7 ln N).