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 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.

We tackle the problem of Byzantine errors in dis-tributed gradient descent within the Byzantine-resilient gradient coding framework. Our proposed solution can recover the exact full gradient in the presence of$s$malicious workers with a data replication factor of only$s$+ 1. It generalizes previous solutions to any data assignment scheme that has a regular replication over all data samples. The scheme detects malicious workers through additional interactive communication and a small number of local computations at the main node, leveraging group-wise comparisons between workers with a provably optimal grouping strategy. The scheme requires at most$s$interactive rounds that incur a total communication cost logarithmic in the number of data samples.

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.