A root-finding algorithm for list decoding of Reed-Muller codes

Let F_q[X₁,...,X_m] denote the set of polynomials over F_q in m variables, and F_q[X₁,...,X_m]_≤u denote the subset that consists of the polynomials of total degree at most u. Let H(T) be a nontrivial polynomial in T with coefficients in F_q[X₁,...,X_m]. A crucial step in interpolation-based list decoding of q-ary Reed-Muller (RM) codes is finding the roots of H(T) in F_q[X₁,...,X_m]_≤u. In this correspondence, we present an efficient root-finding algorithm, which finds all the roots of H(T) in F_q[X₁,...,X_m]_≤u. The algorithm can be used to speed up the list decoding of RM codes.