An efficient transformation method for DFRM expansions

Abstract

Dual Form of Reed-Muller (DFRM) expansions with fixed polarity are derived from Reed-Muller (RM) expansions by using the operation of Kronecker matrix products. An efficient decomposition method is proposed based on the formulation. The method can be used for the transformation between from DFRM expansions to RM expansions within the same fixed polarity as well. Hence, the proposed method is bidirectional. After decomposition, the calculation of the duplicated matrix is avoided, resulting in less computation time. Time complexity of the algorithm is 0(21.5n). The time used for small variables is virtual zero for the tested MCNC benchmarks. For large variable, it still works very well and achieves less than 20 seconds for 25-variable benchmark. In the implementation, only on-set coefficients are used. Consequently, the space complexity is O(M), where M is the number of on-set coefficients. It makes simultaneous optimization in both RM and DFRM expansions possible.