Constructions and Properties of Linear Locally Repairable Codes

TL;DR

This paper investigates the construction and properties of locally repairable codes, presenting methods to modify existing codes, probabilistic proofs for code existence, and explicit constructions of optimal codes over small alphabets.

Contribution

It introduces new methods for modifying codes, probabilistic proofs for code existence, and explicit constructions of optimal vector-linear locally repairable codes.

Findings

Random matrices with added columns can generate locally repairable codes with good minimum distance.

Constructed three infinite classes of optimal vector-linear locally repairable codes.

Provided constructive methods for finding locally repairable codes.

Abstract

In this paper, locally repairable codes with all-symbol locality are studied. Methods to modify already existing codes are presented. Also, it is shown that with high probability, a random matrix with a few extra columns guaranteeing the locality property, is a generator matrix for a locally repairable code with a good minimum distance. The proof of this also gives a constructive method to find locally repairable codes. Constructions are given of three infinite classes of optimal vector-linear locally repairable codes over an alphabet of small size, not depending on the size of the code.