Guruswami, Venkatesan ;
Xing, Chaoping ;
Yuan, Chen
How Long Can Optimal Locally Repairable Codes Be?
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 Singletontype bound dictates the best possible tradeoff between the dimension and distance of LRCs  an LRC attaining this tradeoff is deemed 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 >=slant 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/floor[(d3)/2]}) when d <=slant r+2. Our bounds match when d=5, pinning down n=Theta(q^2) as the asymptotically largest length of an optimal LRC for this case.
BibTeX  Entry
@InProceedings{guruswami_et_al:LIPIcs:2018:9445,
author = {Venkatesan Guruswami and Chaoping Xing and Chen Yuan},
title = {{How Long Can Optimal Locally Repairable Codes Bel}},
booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)},
pages = {41:141:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770859},
ISSN = {18688969},
year = {2018},
volume = {116},
editor = {Eric Blais and Klaus Jansen and Jos{\'e} D. P. Rolim and David Steurer},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9445},
URN = {urn:nbn:de:0030drops94458},
doi = {10.4230/LIPIcs.APPROXRANDOM.2018.41},
annote = {Keywords: Locally Repairable Code, Singleton Bound}
}
2018
Keywords: 

Locally Repairable Code, Singleton Bound 
Seminar: 

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018)

Issue date: 

2018 
Date of publication: 

2018 