The problem of distributed matrix multiplication with straggler tolerance over finite fields is considered, focusing on field sizes for which previous solutions were not applicable (for instance, the field of two elements). We employ Reed-Muller-type codes for explicitly constructing the desired algorithms and study their parameters by translating the problem into a combinatorial problem involving sums of discrete convex sets. We generalize polynomial codes and matdot codes, discussing the impossibility of the latter being applicable for very small field sizes, while providing optimal solutions for some regimes of parameters in both cases.

翻译：本文研究了有限域上具有抗拖尾者容错的分布式矩阵乘法问题，重点关注以往解决方案无法适用的域规模（例如二元域）。我们采用里德-穆勒型编码显式构造目标算法，并通过将该问题转化为涉及离散凸集和的组合问题来研究其参数特性。本文推广了多项式编码与matdot编码，论证了后者在极小域规模下适用的不可能性，同时为两种编码的特定参数区间提供了最优解决方案。