Heights of error-correcting codes

Abstract

In this work, we investigate the evaluation of odd polynomials P defined on a finite field on the class of error-correcting codes C. We exploit the correspondence between codes and Tanner graphs. Thus, we formally define P(C), a polynomial code. Then the new notion of height of a code emerges, whose properties are studied. We extended the lower bound of Tanner on the minimum distance of a code to the case of a polynomial code, by using spectral graph theory. Computer algebra software enable us to give numerical results to illustrate the theory of polynomial codes for various classes of error-correcting codes.