Asymptotic Bounds on the Rate of Locally Repairable Codes

TL;DR

This paper derives improved asymptotic upper bounds on the rate of locally repairable codes with specified distance and locality, especially for linear codes with disjoint repair groups, advancing understanding of code efficiency.

Contribution

It introduces new asymptotic bounds for the rate of linear LRCs with given distance and locality, improving upon previous bounds in the case of disjoint repair groups.

Findings

Bounds are tighter than previous results for linear LRCs.

Disjoint repair groups lead to significant rate improvements.

Applicable over finite fields for codes with linear recovery functions.

Abstract

New asymptotic upper bounds are presented on the rate of sequences of locally repairable codes (LRCs) with a prescribed relative minimum distance and locality over a finite field $F$. The bounds apply to LRCs in which the recovery functions are linear; in particular, the bounds apply to linear LRCs over $F$. The new bounds are shown to improve on previously published results, especially when the repair groups are disjoint, namely, they form a partition of the set of coordinates.