Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes

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Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2019
Major/Subject
Mcode
Degree programme
Language
en
Pages
6089-6100
Series
IEEE Transactions on Information Theory, Volume 65, issue 10
Abstract
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.
Description
Keywords
Maintenance engineering, Linear codes, Upper bound, Optimized production technology, Reliability engineering
Other note
Citation
Grezet , M , Freij-Hollanti , R , Westerbäck , T & Hollanti , C 2019 , ' Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes ' , IEEE Transactions on Information Theory , vol. 65 , no. 10 , 8700214 , pp. 6089-6100 . https://doi.org/10.1109/TIT.2019.2911595