Direct Synthesis Algorithms for Primitive Walsh Function Systems

This paper develops algorithms for the direct synthesis of primitive Walsh systems generated by permutations of the basis functions of the Walsh-Paley mother system by simple Gray codes (SGCs). The algorithms are based on indicator matrices (IMs) of Walsh systems and tables of transitions from the mother system to one or another primitive system. Indicator matrices are n-th order symmetric with respect to the auxiliary diagonal, nondegenerate over , uniquely defining Walsh systems of N-th order, $$( N=2ⁿ) . The PCG subset includes both classical, named left- sided), and alternative (right-sided) codes. The formation of right-sided Gray codes is performed along the direction inverse to the direction of formation of classical codes. In addition to left- and right-sided codes (direct and inverse), the PCG subset includes operators of inverse permutation and cyclic shift of IM strings by one digit to the left or right.