Algebraic Geometry Codes for Distributed Matrix Multiplication Using Local Expansions

TL;DR

This paper introduces algebraic geometry (AG) codes into distributed matrix multiplication, generalizing existing Reed-Solomon based schemes and achieving better recovery thresholds with similar communication costs.

Contribution

It presents the first construction of AG-based PolyDot codes and improves AG-based Polynomial and MatDot codes using local expansions of functions.

Findings

AG-based PolyDot codes are constructed for the first time.

AG-based Polynomial and MatDot codes have better recovery thresholds.

The new basis of the Riemann-Roch space overcomes previous cancellation issues.

Abstract

Code-based Distributed Matrix Multiplication (DMM) has been extensively studied in distributed computing for efficiently performing large-scale matrix multiplication using coding theoretic techniques. The communication cost and recovery threshold (i.e., the least number of successful worker nodes required to recover the product of two matrices) are two major challenges in coded DMM research. Several constructions based on Reed-Solomon (RS) codes are known, including Polynomial codes, MatDot codes, and PolyDot codes. However, these RS-based schemes are not efficient for small finite fields because the distributed order (i.e., the total number of worker nodes) is limited by the size of the underlying finite field. Algebraic geometry (AG) codes can have a code length exceeding the size of the finite field, which helps solve this problem. Some work has been done to generalize Polynomial and MatDot codes to AG codes, but the generalization of PolyDot codes to AGcodes still remains an open problem as far as we know. This is because functions of an algebraic curve do not behave as nicely as polynomials. In this work, by using local expansions of functions, we are able to generalize the three DMM schemes based on RS codes to AG codes. Specifically, we provide a construction of AG-based PolyDot codes for the first time. In addition, our AG-based Polynomial and MatDot codes achieve better recovery thresholds compared to previous AG-based DMM schemes while maintaining similar communication costs. Our constructions are based on a novel basis of the Riemann-Roch space using local expansions, which naturally generalizes the standard monomial basis of the univariate polynomial space in RS codes. In contrast, previous work used the non-gap numbers to construct a basis of the Riemann-Roch space, which can cause cancellation problems that prevent the conditions of PolyDot codes from being satisfied.