  This paper develops an algorithmic approach for obtaining estimates of the
weight enumerators of Reed-Muller (RM) codes. Our algorithm is based on a
technique for estimating the partition functions of spin systems, which in turn
employs a sampler that produces codewords according to a suitably defined Gibbs
distribution. We apply our method to moderate-blocklength RM codes and derive
approximate values of their weight enumerators. We observe that the rates of
the weight enumerator estimates returned by our method are close to the true
rates when these rates are either known or computable by brute-force search; in
other cases, our computations provide provably robust estimates. As a
byproduct, our sampling algorithm also allows us to obtain estimates of the
weight spectra of RM codes. We illustrate our methods by providing estimates of
the hitherto unknown weight enumerators of the RM$(11,5)$ code and the exact
weight spectra of the RM$(10,3)$ and RM$(10,4)$ codes.
