Root-Hadamard transforms and complementary sequences

Abstract

In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, 2k-Hadamard, consta-Hadamard and all HN-transforms. We describe the behavior of what we call the root-Hadamard transform for a generalized Boolean function f in terms of the binary components of f. Further, we define a notion of complementarity (in the spirit of the Golay sequences) with respect to this transform and furthermore, we describe the complementarity of a generalized Boolean set with respect to the binary components of the elements of that set.