Properties and relations for fast linearly independent arithmetic transforms

New fast linearly independent arithmetic (LIA) transforms are introduced here which can be used to represent any functions of binary variables. The transforms are grouped into classes where consistent formulas relating forward and inverse transform matrices are obtained. All the presented transforms have the same computational cost, which is lower than the computational cost of the well-known fixed polarity arithmetic transforms. General classifications and fast forward and inverse transform definitions for all the fast LIA transforms are given. Various properties and mutual relations that exist for the different transforms and their corresponding spectra are also shown. The presented relations and properties reduce the computational cost of finding the best LIA polynomial expansion based on the new transforms.