Distributed matrix multiplication with straggler tolerance over very small field

TL;DR

This paper develops new algorithms for distributed matrix multiplication over very small finite fields, like GF(2), using Reed-Muller-type codes to tolerate stragglers and optimize performance.

Contribution

It introduces Reed-Muller-type codes for small fields, generalizes polynomial and matdot codes, and analyzes their parameters for straggler-tolerant distributed matrix multiplication.

Findings

Reed-Muller-type codes enable matrix multiplication over GF(2).

Optimal solutions are identified for certain parameter regimes.

Matdot codes are shown to be unsuitable for very small fields.

Abstract

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.