Constructions and Properties of Linear Locally Repairable Codes

Abstract

In this paper, locally repairable codes with all-symbol locality are studied. Methods to modify already existing codes are presented. Also, it is shown that with high probability, a random matrix with a few extra columns guaranteeing the locality property, is a generator matrix for a locally repairable code with a good minimum distance. The proof of this also gives a constructive method to find locally repairable codes. Constructions are given of three infinite classes of optimal vector-linear locally repairable codes over an alphabet of small size, not depending on the size of the code.

32 pages. Second code construction in Section V is corrected in this version. Also, some typos are corrected. The results remain the same. Submitted to IEEE Transactions on Information Theory. This is extended, generalized, and clarified version of arXiv:1408.0180

References in corpus (8)

• A family of optimal locally recoverable codes
• Optimal Exact-Regenerating Codes for Distributed Storage at the MSR and MBR Points via a Product-Matrix Construction
• Binary Cyclic Codes that are Locally Repairable
• Codes with Local Regeneration
• MDS Array Codes with Optimal Rebuilding
• Codes between MBR and MSR Points with Exact Repair Property
• Constructions of Optimal and Almost Optimal Locally Repairable Codes
• Self-repairing Homomorphic Codes for Distributed Storage Systems

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• Locally Repairable Codes with Unequal Local Erasure Correction
• Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes
• On minimum distance of locally repairable codes
• On Optimal Ternary Locally Repairable Codes
• Constructions of Optimal Cyclic $(r,\delta )$ Locally Repairable Codes
• On Binary Matroid Minors and Applications to Data Storage over Small Fields
• Optimal Locally Repairable Codes with Improved Update Complexity
• Codes with Unequal Disjoint Local Erasure Correction Constraints
• Locally Repairable Codes with Multiple $(r_i,{\delta }_i)$-Localities