Distributed Matrix Multiplication-Friendly Algebraic Function Fields
Yun Long Zhu, Chang-An Zhao
TL;DR
This work tackles efficient distributed matrix multiplication over finite fields by introducing DMM-friendly algebraic function fields tailored for polynomial AG codes and Matdot AG codes. The authors prove optimal recovery-threshold bounds $R \ge g(F) + mn$ for polynomial AG codes and $R \ge 2g(F) + 2m - 1$ for Matdot AG codes (when $m \ge g(F)+1$), and construct explicit function-field families achieving these thresholds through extensions of the rational function field. They develop polynomial-code- and Matdot-code-friendly field constructions, analyze decoding complexity, and provide implementation results showing practical speedups over rational-field baselines and improved straggler tolerance. The work highlights iterative opportunities to minimize genus and enhance field practicality, and points toward extensions to locally repairable AG codes and related DMM frameworks.
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.