How long can optimal locally repairable codes be?

TL;DR

This paper establishes bounds on the maximum length of optimal locally repairable codes (LRCs) over fixed alphabets for various code distances, revealing that for distances $d \,\ge\, 5$, the length is tightly constrained by the alphabet size.

Contribution

The authors prove upper bounds on the length of optimal LRCs for distances $d \,\ge\, 5$ and construct codes that match these bounds for specific parameters, clarifying the maximum achievable length.

Findings

Optimal LRC length is at most $O(d q^3)$ for $d \,\ge\, 5$.

For $d=5$, the maximum length is $O(q^2)$.

Existence of optimal LRCs with length $\,\Omega_{d,r}(q^{1+1/\lfloor(d-3)/2\rfloor})$ when $d \le r+2$.

Abstract

A locally repairable code (LRC) with locality $r$ allows for the recovery of any erased codeword symbol using only $r$ other codeword symbols. A Singleton-type bound dictates the best possible trade-off between the dimension and distance of LRCs --- an LRC attaining this trade-off is deemed \emph{optimal}. Such optimal LRCs have been constructed over alphabets growing linearly in the block length. Unlike the classical Singleton bound, however, it was not known if such a linear growth in the alphabet size is necessary, or for that matter even if the alphabet needs to grow at all with the block length. Indeed, for small code distances $3,4$, arbitrarily long optimal LRCs were known over fixed alphabets. Here, we prove that for distances $d \ge 5$, the code length $n$ of an optimal LRC over an alphabet of size $q$ must be at most roughly $O(d q^3)$. For the case $d=5$, our upper bound is $O(q^2)$. We complement these bounds by showing the existence of optimal LRCs of length $\Omega_{d,r}(q^{1+1/\lfloor(d-3)/2\rfloor})$ when $d \le r+2$. These bounds match when $d=5$, thus pinning down $n=\Theta(q^2)$ as the asymptotically largest length of an optimal LRC for this case.