Optimal repair of reed-solomon codes: Achieving the cut-set bound

Itzhak Tamo*, Min Ye, Alexander Barg

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

42 Scopus citations

Abstract

The repair problem for an (n, k) error-correcting code calls for recovery of an unavailable coordinate of the codeword by downloading as little information as possible from a subset of the remaining coordinates. Using the terminology motivated by coding in distributed storage, we attempt to repair a failed node by accessing information stored on d helper nodes, where k ≪ d ≪ n - 1, and using as little repair bandwidth as possible to recover the lost information.By the so-called cut-set bound (Dimakis et al., 2010), the repair bandwidth of an (n,k = n - r) MDS code using d helper nodes is at least dl/(d + 1 - k), where l is the size of the node. A number of constructions of MDS array codes have been shown to meet this bound with equality. In a related but separate line of work, Guruswami and Wootters (2016) studied repair of Reed-Solomon (RS) codes, showing that it is possible to perform repair using a smaller bandwidth than under the trivial approach. At the same time, their work as well as follow-up papers stopped short of constructing RS codes (or any scalar MDS codes) that meet the cut-set bound with equality, which has been an open problem in coding theory.In this work we present a solution to this problem, constructing RS codes of length n over the field of size (ql, l = exp((1 + o(1)n log n) that meet the cut-set bound. We also prove an almost matching lower bound on l, showing that super-exponential scaling is both necessary and sufficient for achieving the cut-set bound using linear repair schemes. More precisely, we prove that for scalar MDS codes (including the RS codes) to meet this bound, the sub-packetization l must satisfy l ≫ exp((1 + o(1))k log k).

Original languageEnglish
Title of host publicationProceedings - 58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017
PublisherIEEE Computer Society
Pages216-227
Number of pages12
ISBN (Electronic)9781538634646
DOIs
StatePublished - 10 Nov 2017
Event58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017 - Berkeley, United States
Duration: 15 Oct 201717 Oct 2017

Publication series

NameAnnual Symposium on Foundations of Computer Science - Proceedings
Volume2017-October
ISSN (Print)0272-5428

Conference

Conference58th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2017
Country/TerritoryUnited States
CityBerkeley
Period15/10/1717/10/17

Funding

FundersFunder number
National Science Foundation
Directorate for Computer and Information Science and Engineering1422955
Directorate for Computer and Information Science and Engineering

    Keywords

    • Cut-set bound
    • Optimal sub-packetization
    • Repair bandwidth

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