On the Single-Parity Locally Repairable Codes

Yanbo LU  Jie HAO  Shu-Tao XIA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E100-A   No.6   pp.1342-1345
Publication Date: 2017/06/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E100.A.1342
Type of Manuscript: LETTER
Category: Coding Theory
linear codes,  erasure codes,  code locality,  locally repairable codes,  distance bounds,  

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Locally repairable codes (LRCs) have attracted much interest recently due to their applications in distributed storage systems. In an [n,k,d] linear code, a code symbol is said to have locality r if it can be repaired by accessing at most r other code symbols. An (n,k,r) LRC with locality r for the information symbols has minimum distance dn-k-⌈k/r⌉+2. In this letter, we study single-parity LRCs where every repair group contains exactly one parity symbol. Firstly, we give a new characterization of single-parity LRCs based on the standard form of generator matrices. For the optimal single-parity LRCs meeting the Singleton-like bound, we give necessary conditions on the structures of generator matrices. Then we construct all the optimal binary single-parity LRCs meeting the Singleton-like bound dn-k-⌈k/r⌉+2.