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.

翻译：本文开发了一种算法方法，用于获得Reed-Muller（RM）码重量枚举子的估计值。我们的算法基于自旋系统配分函数的估计技术，该技术利用一个采样器，根据适当定义的吉布斯分布生成码字。我们将该方法应用于中等码长的RM码，并推导出其重量枚举子的近似值。我们观察到，当这些速率已知或可通过暴力搜索计算时，我们的方法返回的重量枚举子估计速率接近真实速率；在其他情况下，我们的计算提供了可证明的稳健估计。作为副产品，我们的采样算法还使我们能够获得RM码重量谱的估计值。我们通过提供迄今未知的RM$(11,5)$码重量枚举子的估计值，以及RM$(10,3)$和RM$(10,4)$码的精确重量谱，来说明我们的方法。