On Pivot Orbits of Boolean Functions

Abstract

We derive a spectral interpretation of the pivot operation on a graph and generalise this operation to hypergraphs. We establish lower bounds on the number of flat spectra of a Boolean function, depending on internal structures, with respect to the {I,H}^n and {I,H,N}^n sets of transforms. We also construct a family of Boolean functions of degree higher than two with a large number of flat spectra with respect to {I,H}^n, and compute a lower bound on this number. The relationship between pivot orbits and equivalence classes of error-correcting codes is then highlighted. Finally, an enumeration of pivot orbits of various types of graphs is given, and it is shown that the same technique can be used to classify codes.