Distributed Matrix Multiplication-Friendly Algebraic Function Fields

TL;DR

This paper develops specialized algebraic function fields that facilitate distributed matrix multiplication with optimal recovery thresholds, improving practical implementation despite some efficiency trade-offs.

Contribution

It introduces new DMM-friendly algebraic function fields supporting optimal recovery thresholds for polynomial and Matdot codes, with explicit constructions and examples.

Findings

Supports DMM with optimal recovery thresholds

Provides explicit constructions of algebraic function fields

Offers practical improvements for matrix multiplication implementations

Abstract

In this paper, we introduce distributed matrix multiplication (DMM)-friendly algebraic function fields for polynomial codes and Matdot codes, and present several constructions for such function fields through extensions of the rational function field. The primary challenge in extending polynomial codes and Matdot codes to algebraic function fields lies in constructing optimal decoding schemes. We establish optimal recovery thresholds for both polynomial algebraic geometry (AG) codes and Matdot AG codes for fixed matrix multiplication. Our proposed function fields support DMM with optimal recovery thresholds, while offering rational places that exceed the base finite field size in specific parameter regimes. Although these fields may not achieve optimal computational efficiency, our results provide practical improvements for matrix multiplication implementations. Explicit examples of applicable function fields are provided.