On the Distance Distribution of Reed-Muller Codes

In this paper, we give error bounds for the distance distribution of Reed-Muller codes, extending prior work on the distance distribution of Reed-Solomon codes. This is equivalent to the problem of counting multivariate polynomials over a finite field with prescribed degree, coefficients, and number of zeroes. We provide a solution to this problem using the character sum method, which offers a new unified framework applicable to a broad class of polynomial enumeration problems over finite fields that involve prescribed evaluation vectors. This work effectively makes the first systematic attempt to study the coset weight distribution problem for Reed-Muller codes of fixed degree over large finite fields, which was proposed in MacWilliams and Sloane's 1977 textbook \emph{The Theory of Error Correcting Codes}.