Abstract
Locally repairable codes (LRCs) are a new family of erasure codes used in distributed storage systems which have attracted a great deal of interest in recent years. For an [ n , k , d ] linear code, if a code symbol can be repaired by t disjoint groups of other code symbols, where each group contains at most r code symbols, it is said to have availability- ( r , t ) . Single-parity LRCs are LRCs with a constraint that each repairable group contains exactly one parity symbol. For an [ n , k , d ] single-parity LRC with availability- ( r , t ) for the information symbols (single-parity LRCs), the minimum distance satisfies d ≤ n − k − ⌈ k t / r ⌉ + t + 1 . In this paper, we focus on the study of single-parity LRCs with availability- ( r , t ) for information symbols. Based on the standard form of generator matrices, we present a novel characterization of single-parity LRCs with availability t ≥ 1 . Then, a simple and straightforward proof for the Singleton-type bound is given based on the new characterization. Some necessary conditions for optimal single-parity LRCs with availability t ≥ 1 are obtained, which might provide some guidelines for optimal coding constructions.
Highlights
In the era of big data, more and more modern large-scale distributed storage systems (DSSs) tend to use erasure codes as the redundancy scheme rather than simple replication redundancy schemes [1,2]
While here we focus on Locally repairable codes (LRCs) with multiple reparable groups and study the general case of availability t ≥ 1
We give the introduction of replication schemes, erasure codes, and we focus on LRCs with parameters (n, k, r, t)
Summary
In the era of big data, more and more modern large-scale distributed storage systems (DSSs) tend to use erasure codes as the redundancy scheme rather than simple replication redundancy schemes [1,2]. RS codes can achieve great improvements on the storage performance of DSSs compared with replication schemes, they suffer from high repair cost for a failed storage node. We can see that the repair cost of conventional RS codes depends on the code dimension k, while the locality r of LRCs improves the repair cost of failed storage blocks. In. Section 4, we study single-parity LRCs on the basis of the generator matrices with the standard form under the condition of availability t ≥ 1, and give the novel proof of the bound (3) and some necessary conditions of optimal codes.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
