Bounds on the Size of Locally Recoverable Codes

• Published in: IEEE transactions on information theory 2015-11, Vol.61 (11), p.5787-5794
• Main Authors: Cadambe, Viveck R., Mazumdar, Arya
• Format: Article
• Language: English
• Subjects: Binary codes; Codes; Coding; Concatenated codes; Constants; Information theory; Linear codes; Maintenance engineering; Mathematical models; Optimization; Parity check codes; Symbols; Upper bound
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2015.2477406

In a locally recoverable or repairable code, any symbol of a codeword can be recovered by reading only a small (constant) number of other symbols. The notion of local recoverability is important in the area of distributed storage where a most frequent error-event is a single storage node failure (erasure). A common objective is to repair the node by downloading data from as few other storage nodes as possible. In this paper, we bound the minimum distance of a code in terms of its length, size, and locality. Unlike the previous bounds, our bound follows from a significantly simple analysis and depends on the size of the alphabet being used. It turns out that the binary Simplex codes satisfy our bound with equality; hence, the Simplex codes are the first example of an optimal binary locally repairable code family. We also provide achievability results based on random coding and concatenated codes that are numerically verified to be close to our bounds.