Multiple-valued Logic Functions and Error-correcting Codes

Abstract

A multiple-valued logic function is a multivariable function whose input and output are in a finite field and is applied to the construction of switching circuits. A multiple-valued logic polynomial is a multivariable polynomial with coefficients in a finite field. It is known that multiple-valued logic functions are equal to multiple-valued logic polynomials. Because error-correcting codes and multiple-valued logic polynomials have a dual relationship with each other, the research results of error-correcting codes can be employed to multiple-valued logic polynomials. In this study, a generalized “convolution theorem” is described, which is a relationship between multiple-valued logic functions and polynomials through discrete Fourier transforms, and it is generalized to a subset of finite fields called a semigroup. Next, this theorem is applied to speed up the multiplication of multiple-valued logic polynomials. Finally, to show a relationship between multiple-valued logic functions and error-correcting codes, a method of explicitly obtaining dual codes is given.