Classification and properties of fast linearly independent logic transformations

Properties of fastest linearly independent transforms over GF(3)

“…They are of great interest because the Reed-Muller based circuits have many desirable properties, such as simple testability vectors. The binary Reed-Muller transform itself is only one type of Linearly Independent (LI) transforms introduced in [7] and discussed in detail in [4] that are the special case of more general orthonormal expansions discussed in [1]. Further research showed that there are some LI transforms having the smallest computational cost that is lower than that of the binary Reed-Muller transform [8].…”

Properties and Relations of Ternary Linearly Independent Transforms

“…Linearly independent logic has proved to be not only of great theoretical value, but also of practical value, to the development of new types of decision diagrams and design of fine-grain and cellular automata types of Field Programmable Gate Arrays (FPGAs) and different Programmable Logic Devices (PLDs) with XOR gates [7]. Another important development in further consideration of different linearly independent functions as the efficient tools in the designing of modern FPGAs and PLDs was the identification of those linearly independent expansions for which there exist fast forward and inverse transformation and for efficient calculation of their spectral coefficients [4] and hence finding the final hardware implementations.…”

“…The Shannon, Reed-Muller and related spectral expansions used in logic synthesis, analysis, and testing were discussed by many authors [1][2][3][4][5][6]. It is important to notice that Shannon's expansion is a special case of an expansion using an orthonormal basis based on generalized cofactors that may not be unique [3].…”

Fast transforms for multiple-valued input binary output PLI logic

“…Some of its properties were discussed in [4,5,[10][11][12]. A theorem from [10] includes all binary logic circuits that are realized in GF(2) algebra.…”

“…PLI logic is extremely well suited to implementations in the form of fine grain FPGAs [10]. The theory has been developed that allows one to find and calculate optimal PLI expansions by using fast transforms [4,5]. For a selected class of PLI Transformation, the corresponding polynomial expansion of an arbitrary logical function is canonical.…”