Bounds on the Rate of Linear Locally Repairable Codes over Small Alphabets

TL;DR

This paper establishes the tightest known upper bound on the rate of linear locally repairable codes over small alphabets, advancing understanding of their fundamental limits in distributed storage applications.

Contribution

It provides the most precise upper bound on the rate-distance trade-off for linear LRCs over small alphabets, improving upon previous bounds.

Findings

Derived the tightest known upper bound on the rate of linear LRCs

Enhanced understanding of the rate-distance trade-off for small alphabet codes

Improved previous bounds such as those in Cadambe et al. (2013)

Abstract

Locally repairable codes (LRC) have recently been a subject of intense research due to theoretical appeal and their application in distributed storage systems. In an LRC, any coordinate of a codeword can be recovered by accessing only few other coordinates. For LRCs over small alphabet (such as binary), the optimal rate-distance trade-off is unknown. In this paper we provide the tightest known upper bound on the rate of linear LRCs of a given relative distance, an improvement over any previous result, in particular \cite{cadambe2013upper}.