A numerical method for parallel calculation of the positional characteristic for error correction in a polyalphabetic polynomial modular code

Abstract

The trend toward increasing the efficiency of computing systems and devices is directly related with the transition to parallel computing. We propose that parallel calculations should be conducted at the level of arithmetic operations using arithmetic composite modular codes (CMC), in which code combinations represent a set of residues obtained by dividing an integer into bases. There are two types of such codes. In the polyalphabetic code of the residual number system (PCRNS), mutually prime numbers are used as bases. In polyalphabetic polynomial modular code (PPMC) there are irreducible polynomials. A characteristic feature of these codes is that addition, subtraction and multiplication operations are implemented in parallel in bases. There is no data exchange between the bases. As a result, an increase in the productivity of computing systems is achieved. The bases of polyalphabetic modular codes are equal, independent, and serve as a basis for constructing arithmetic codes that detect and correct errors that occur in the calculation process. The article presents theoretical foundations for constructing a redundant PPMC capable of detecting and correcting computational errors. On the basis of the proved theorems, a numerical method for calculating the positional characteristic (PH) of the polynomial interval in PPMC was developed. This method requires fewer multiplication operations compared to the classical method of calculating this PH. Examples of the application of this method are considered.