ABSTRACT

The Gray transformations are interpreted in the work as a generalization of concept of the Gray codes. In the well-known (classical) schemes, the process of forming the direct and reverse Gray codes develops from left to right. At the same time, the most significant (left) digit of the number being transformed does not change under both direct and inverse transformations. Apparently, they turned out to be out of sight of both mathematicians and hardware developers the possibility of constructing codes that are inverse in the direction of the formation of the classical Gray codes. The article proposed a method of transformation, called the right-sided transformation, which is inverse in the direction of the formation of the classical Gray code. In the new class of right-sided transformations, with the direct and inverse transformations, the value of the lower (right) digit of the number being converted remains unchanged. The combination of left and right Gray conversion (both direct and inverse) together with the inverse permutation of codes led to the possibility of constructing combined or composite Gray codes. The use of composite Gray codes turned out to be very successful in problems of determining the structure and interrelation of symmetric systems of Walsh functions, discrete the Vilenkin-Crestenson functions, in cryptography, coding, and in other applications. Original systems of the classical Walsh–Cooley functions and Walsh–Tukey sequential functions are developed, possessing unique properties, which, being used as bases in DFT processors, are the only Walsh systems that deliver linear connectivity to the frequency scales of processors for input discrete complex exponential signals.