Remarks on Codes, Spectral Transforms, and Decision Diagrams

In this paper, we discuss definitions, features, and relationships of Reed-Muller transforms, Reed-Muller codes and their generalizations to multiple-valued cases, and Reed-Muller decision diagrams. The novelty in this primarily review paper resides in putting together these concepts in the same context and providing a uniform point of view to their definition in terms of a convolutionwise multiplication. In particular, we point out that the Plotkin construction schemes for Reed-Muller codes used in coding theory are a different notation for basic Reed-Muller transform matrices over finite fields or can be alternatively viewed as decomposition rules used to define the Reed-Muller decision diagrams.