Algebraic Geometric Rook Codes for Coded Distributed Computing

TL;DR

This paper introduces algebraic geometric codes for distributed matrix multiplication over finite fields, enabling fault-tolerant computation with more workers than field size, generalizing prior results and linking recovery thresholds to function field genus.

Contribution

It extends coded distributed computing to finite fields with more workers than field size using algebraic geometric codes, achieving fault tolerance and generalizing previous work.

Findings

Codes achieve recovery thresholds proportional to matrix multiplication complexity

All functions can be computed fault-tolerantly over finite fields

Recovery threshold related to the genus of the underlying function field

Abstract

We extend coded distributed computing over finite fields to allow the number of workers to be larger than the field size. We give codes that work for fully general matrix multiplication and show that in this case we serendipitously have that all functions can be computed in a distributed fault-tolerant fashion over finite fields. This generalizes previous results on the topic. We prove that the associated codes achieve a recovery threshold similar to the ones for characteristic zero fields but now with a factor that is proportional to the genus of the underlying function field. In particular, we have that the recovery threshold of these codes is proportional to the classical complexity of matrix multiplication by a factor of at most the genus.