# Near-optimal Repair of Reed-Solomon Codes with Low Sub-packetization

## TL;DR

This paper introduces a new class of Reed-Solomon codes called $psilon$-MSR codes that achieve near-optimal repair bandwidth with polynomial sub-packetization, improving practicality over previous exponential solutions.

## Contribution

It constructs constant-rate $psilon$-MSR Reed-Solomon codes with polynomial sub-packetization, balancing repair bandwidth and sub-packetization size.

## Key findings

- Polynomial sub-packetization achieved for $psilon$-MSR codes.
- Explicit tradeoff between repair bandwidth and sub-packetization.
- Improved practicality over exponential sub-packetization codes.

## Abstract

Minimum storage regenerating (MSR) codes are MDS codes which allow for recovery of any single erased symbol with optimal repair bandwidth, based on the smallest possible fraction of the contents downloaded from each of the other symbols. Recently, certain Reed-Solomon codes were constructed which are MSR. However, the sub-packetization of these codes is exponentially large, growing like $n^{\Omega(n)}$ in the constant-rate regime. In this work, we study the relaxed notion of $\epsilon$-MSR codes, which incur a factor of $(1+\epsilon)$ higher than the optimal repair bandwidth, in the context of Reed-Solomon codes. We give constructions of constant-rate $\epsilon$-MSR Reed-Solomon codes with polynomial sub-packetization of $n^{O(1/\epsilon)}$ and thereby giving an explicit tradeoff between the repair bandwidth and sub-packetization.