Close Encounters with Boolean Functions of Three Different Kinds

Complex arrays with good aperiodic properties are characterised and it is shown how the joining of dimensions can generate sequences which retain the aperiodic properties of the parent array. For the case of 2× 2× . . .× 2 arrays we define two new notions of aperiodicity by exploiting a unitary matrix represention. In particular, we apply unitary rotations by members of a size-3 cyclic subgroup of the local Clifford group to the aperiodic description. It is shown how the three notions of aperiodicity relate naturally to the autocorrelations described by the action of the Heisenberg-Weyl group. Finally, after providing some cryptographic motivation for two of the three aperiodic descriptions, we devise new constructions for complementary pairs of Boolean functions of three different kinds, and give explicit examples for each.