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

Matthias Grezet, Ragnar Freij-Hollanti, Thomas Westerbäck, Camilla Hollanti

Research output: Contribution to journalArticleScientificpeer-review

6 Citations (Scopus)
174 Downloads (Pure)


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.
Original languageEnglish
Article number8700214
Pages (from-to)6089-6100
Number of pages12
JournalIEEE Transactions on Information Theory
Issue number10
Publication statusPublished - 2019
MoE publication typeA1 Journal article-refereed


  • Maintenance engineering
  • Linear codes
  • Upper bound
  • Optimized production technology
  • Reliability engineering


Dive into the research topics of 'Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes'. Together they form a unique fingerprint.

Cite this