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Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes
Aalto Univ, Dept Math & Syst Anal, Aalto 00076, Finland..ORCID iD: 0000-0001-7612-5967
Aalto Univ, Dept Math & Syst Anal, Aalto 00076, Finland..
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. (MAM)
Aalto Univ, Dept Math & Syst Anal, Aalto 00076, Finland..
2019 (English)In: IEEE Transactions on Information Theory, ISSN 0018-9448, E-ISSN 1557-9654, Vol. 65, no 10, p. 6089-6100, article id 8700214Article in journal (Refereed) Published
Abstract [en]

Locally repairable codes (LRCs) have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others. Several works have thus been carried out to understand the optimal rate-distance tradeoff, but only recently the size of the alphabet has been taken into account. In this paper, a novel definition of locality is proposed to keep track of the precise number of nodes required for a local repair when the repair sets do not yield MDS codes. Then, a new alphabet-dependent bound is derived, which applies both to the new definition and the initial definition of locality. The new bound is based on consecutive residual codes and intrinsically uses the Griesmer bound. A special case of the bound yields both the extension of the Cadambe-Mazumdar bound and the Singleton-type bound for codes with locality (r, delta), implying that the new bound is at least as good as these bounds. Furthermore, an upper bound on the asymptotic rate-distance tradeoff of LRCs is derived, and yields the tightest known upper bound for large relative minimum distances. Achievability results are also provided by deriving the locality of the family of Simplex codes together with a few examples of optimal codes.

Place, publisher, year, edition, pages
IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC , 2019. Vol. 65, no 10, p. 6089-6100, article id 8700214
Keywords [en]
Locally repairable codes (LRCs), upper bound, Griesmer bound
National Category
Computational Mathematics Discrete Mathematics Algebra and Logic Other Electrical Engineering, Electronic Engineering, Information Engineering
Research subject
Mathematics/Applied Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-46332DOI: 10.1109/TIT.2019.2911595ISI: 000487041200008Scopus ID: 2-s2.0-85077357211OAI: oai:DiVA.org:mdh-46332DiVA, id: diva2:1378192
Available from: 2019-12-13 Created: 2019-12-13 Last updated: 2020-01-16Bibliographically approved

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Westerback, Thomas

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