Distributed matrix multiplication with straggler tolerance over very small field

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.