# Collaborative Decoding of Polynomial Codes for Distributed Computation

## TL;DR

This paper demonstrates that polynomial codes used in distributed matrix multiplication are interleaved Reed-Solomon codes, enabling collaborative decoding to correct errors and improve numerical stability in distributed computation systems.

## Contribution

It reveals the interleaved Reed-Solomon structure of polynomial codes and introduces a collaborative decoding approach for fault-tolerant distributed matrix multiplication.

## Key findings

- Errors can be corrected with probability 1 when t < N-K under Gaussian noise.
- Collaborative decoding improves numerical stability as the number of Reed-Solomon codes increases.
- The approach enhances fault tolerance in distributed matrix multiplication systems.

## Abstract

We show that polynomial codes (and some related codes) used for distributed matrix multiplication are interleaved Reed-Solomon codes and, hence, can be collaboratively decoded. We consider a fault tolerant setup where $t$ worker nodes return erroneous values. For an additive random Gaussian error model, we show that for all $t < N-K$, errors can be corrected with probability 1. Further, numerical results show that in the presence of additive errors, when $L$ Reed-Solomon codes are collaboratively decoded, the numerical stability in recovering the error locator polynomial improves with increasing $L$.