Generation and properties of new fastest linearly independent transforms over GF(2) with reordering

Several fastest linearly independent (LI) transforms over GF(2) as well as their properties have been presented in recent papers. The transforms are able to provide more effective representations than the Reed-Muller transform for some binary functions. In this paper, new fastest LI transforms are introduced which are obtained by multiplying the factorized transform matrices of the previously defined fastest LI transforms in different orderings. This way of generation ensures that the new transforms have the same computational costs as the original fastest LI transforms and that they possess fast forward and inverse transforms. Properties of the new transforms are also investigated and their experimental results for some binary benchmark functions are given. From comparison of experimental results, it is shown that the new transforms are useful as they are able to give polynomial expansions with smaller number of nonzero spectral coefficients than those of the original fastest LI transforms for most binary benchmark functions.