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# Construction of Optimal Locally Recoverable Codes and Connection with Hypergraph

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LIPIcs.ICALP.2019.98.pdf
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## Acknowledgements

We sincerely thank Prof. J. Verstraëte for his linking our condition (8) with the problem in extremal graph theory. He also provided us some references for latest results on extremal graph theory. We would also like to express our great gratitude to Profs. V. Guruswami, Q. Xiang and M. Lu for discussions and help.

## Cite As

Chaoping Xing and Chen Yuan. Construction of Optimal Locally Recoverable Codes and Connection with Hypergraph. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 98:1-98:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ICALP.2019.98

## Abstract

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].

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Error-correcting codes
##### Keywords
• Locally Repairable Codes
• Hypergraph

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## References

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