Eigenspaces of the discrete Walsh transform

We refine the notion of a discrete Walsh function and generalize the notion of a discrete Walsh transform, for which we propose a method for generating a corresponding $W$-matrix. We propose spectral decompositions of the discrete Walsh transform operators in arbitrary enumerations, as well as methods for finding bases of eigenspaces, one of them using a new direct product of matrices. We propose a notation for the fast discrete Walsh transform algorithm in the Paley enumeration. We construct Parseval frames for eigenspaces of the discrete Walsh transform in the Paley enumeration and demonstrate methods for applying them in error detection and correction.