The problem of straggler mitigation in distributed matrix multiplication (DMM) is considered for a large number of worker nodes and a fixed small finite field. Polynomial codes and matdot codes are generalized by making use of algebraic function fields (i.e., algebraic functions over an algebraic curve) over a finite field. The construction of optimal solutions is translated to a combinatorial problem on the Weierstrass semigroups of the corresponding algebraic curves. Optimal or almost optimal solutions are provided. These have the same computational complexity per worker as classical polynomial and matdot codes, and their recovery thresholds are almost optimal in the asymptotic regime (growing number of workers and a fixed finite field).

翻译：考虑在大量工作节点和固定小有限域场景下的分布式矩阵乘法（DMM）中的掉队节点缓解问题。通过利用有限域上的代数函数域（即代数曲线上的代数函数），将多项式编码和Matdot编码进行了推广。最优解的构造被转化为对应代数曲线上Weierstrass半群的组合问题。本文提供了最优或接近最优的解决方案，这些方案在每个工作节点上的计算复杂度与经典多项式编码和Matdot编码相同，且其恢复阈值在渐近意义下（工作节点数增长且有限域固定）接近最优。