Sampling-Based Estimates of the Weight Enumerators of Reed-Muller Codes

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 by-product, our sampling algorithm also allows us to put together the weight spectrum, i.e., the weights at which the enumerators are non-zero, of an RM code, by providing witnesses in the form of codewords at each weight in the spectrum. We illustrate our method by providing estimates of the hitherto unknown weight enumerators of the RM $(11,5)$ code for weights that are multiples of 4 between 512 and 1024. We also obtain the exact weight spectrum of the RM $(10,4)$ code.