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As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the research of LRCs, bounds and constructions of the corresponding optimal codes are of particular concerns. In this work, codes with $(r,delta)$-locality which have optimal minimal distance w.r.t. the bound given by Prakash et al. cite{Prakash2012Optimal} are considered. Through parity check matrix approach, constructions of both optimal $(r,delta)$-LRCs with all symbol locality ($(r,delta)_a$-LRCs) and optimal $(r,delta)$-LRCs with information locality ($(r,delta)_i$-LRCs) are provided. As a generalization of a work of Xing and Yuan cite{XY19}, these constructions are built on a connection between sparse hypergraphs and optimal $(r,delta)$-LRCs. With the help of constructions of large sparse hypergraphs, the length of codes constructed can be super-linear in the alphabet size. This improves upon previous constructions when the minimal distance of the code is at least $3delta+1$. As two applications, optimal H-LRCs with super-linear length and GSD codes with unbounded length are also constructed.
Locally repairable codes with locality $r$ ($r$-LRCs for short) were introduced by Gopalan et al. cite{1} to recover a failed node of the code from at most other $r$ available nodes. And then $(r,delta)$ locally repairable codes ($(r,delta)$-LRCs for
This paper aims to go beyond resilience into the study of security and local-repairability for distributed storage systems (DSS). Security and local-repairability are both important as features of an efficient storage system, and this paper aims to u
In this work it is shown that locally repairable codes (LRCs) can be list-decoded efficiently beyond the Johnson radius for a large range of parameters by utilizing the local error-correction capabilities. The corresponding decoding radius is derived
We consider error decoding of locally repairable codes (LRC) and partial MDS (PMDS) codes through interleaved decoding. For a specific class of LRCs we investigate the success probability of interleaved decoding. For PMDS codes we show that there is
We introduce a new family of Fountain codes that are systematic and also have sparse parities. Given an input of $k$ symbols, our codes produce an unbounded number of output symbols, generating each parity independently by linearly combining a logari