Locally recoverable codes are a class of block codes with an additional property called locality. A locally recoverable code with locality r can recover a symbol by reading at most r other symbols. Recently, it was discovered by several authors that a q-ary optimal locally recoverable code, i.e., a locally recoverable code achieving the Singleton-type bound, can have length much bigger than q+1. In this paper, we present both the upper bound and the lower bound on the length of optimal locally recoverable codes. Our lower bound improves the best known result in [Yuan Luo et al., 2018] for all distance d >= 7. This result is built on the observation of the parity-check matrix equipped with the Vandermonde structure. It turns out that a parity-check matrix with the Vandermonde structure produces an optimal locally recoverable code if it satisfies a certain expansion property for subsets of F_q. To our surprise, this expansion property is then shown to be equivalent to a well-studied problem in extremal graph theory. Our upper bound is derived by an refined analysis of the arguments of Theorem 3.3 in [Venkatesan Guruswami et al., 2018].
@InProceedings{xing_et_al:LIPIcs.ICALP.2019.98, author = {Xing, Chaoping and Yuan, Chen}, title = {{Construction of Optimal Locally Recoverable Codes and Connection with Hypergraph}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {98:1--98:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.98}, URN = {urn:nbn:de:0030-drops-106745}, doi = {10.4230/LIPIcs.ICALP.2019.98}, annote = {Keywords: Locally Repairable Codes, Hypergraph} }
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