Sampling-Based Estimates of the Sizes of Constrained Subcodes of Reed-Muller Codes

Abstract: This paper develops an algorithmic approach for obtaining approximate, numerical estimates of the sizes of subcodes of Reed-Muller (RM) codes, all of the codewords in which satisfy a given constraint. Our algorithm is based on a statistical physics technique for estimating the partition functions of spin systems, which in turn makes use of a sampler that produces RM codewords according to a Gibbs distribution. The Gibbs distribution is designed so that it is biased towards codewords that respect the constraint. We apply our method to approximately compute the sizes of runlength limited (RLL) subcodes and obtain estimates of the weight distribution of moderate-blocklength RM codes. We observe that the estimates returned by our method are close to the true sizes when these sizes are either known or computable by brute-force search; in other cases, our computations provide provably robust estimates. As an illustration of our methods, we provide estimates of the weight distribution of the RM$(9,4)$ code.