New enumeration of Walsh matrices

The discrete Walsh transform is a linear transform defined by a Walsh matrix. Three ways to construct Walsh matrices are known, which differ by the sequence order of rows and correspond to the Paley, Walsh, and Hadamard enumerations. We propose a new enumeration of Walsh matrices and study its properties. The new enumeration is constructed as a linear rearrangement; we obtain an eigenvector basis for it and propose a convenient-to-generate fast implementation algorithm; the new enumeration possesses certain symmetry properties, which make it similar to the discrete Fourier transform.