# An Exponential Lower Bound on the Sub-Packetization of MSR Codes

## TL;DR

This paper proves an exponential lower bound on the sub-packetization size of MSR codes, establishing a fundamental limit that matches known upper bounds and resolving a key open problem in distributed storage coding theory.

## Contribution

It provides an almost tight exponential lower bound on the sub-packetization of MSR codes, settling a major open question in the field.

## Key findings

- Lower bound: ( ext{k/r}) on sub-packetization size ( ext{ell}) for MSR codes
- Previous bounds were ( ext{exp}( ext{sqrt}(k/r)))
- The result matches known upper bounds, confirming the exponential growth requirement.

## Abstract

An $(n,k,\ell)$-vector MDS code is a $\mathbb{F}$-linear subspace of $(\mathbb{F}^\ell)^n$ (for some field $\mathbb{F}$) of dimension $k\ell$, such that any $k$ (vector) symbols of the codeword suffice to determine the remaining $r=n-k$ (vector) symbols. The length $\ell$ of each codeword symbol is called the sub-packetization of the code. Such a code is called minimum storage regenerating (MSR), if any single symbol of a codeword can be recovered by downloading $\ell/r$ field elements (which is known to be the least possible) from each of the other symbols. MSR codes are attractive for use in distributed storage systems, and by now a variety of ingenious constructions of MSR codes are available. However, they all suffer from exponentially large sub-packetization $\ell \gtrsim r^{k/r}$. Our main result is an almost tight lower bound showing that for an MSR code, one must have $\ell \ge \exp(\Omega(k/r))$. This settles a central open question concerning MSR codes that has received much attention. Previously, a lower bound of $\approx \exp(\sqrt{k/r})$, and a tight lower bound for a restricted class of "optimal access" MSR codes, were known.