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Construction of Optimal Locally Recoverable Codes and Connection with Hypergraph

Authors Chaoping Xing , Chen Yuan

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

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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].

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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

Author Details

Chaoping Xing
  • School of Electronics, Information and Electrical Engineering, Shanghai Jiaotong University, Shanghai, 200240, P. R. China
Chen Yuan
  • Centrum Wiskunde & Informatica, Amsterdam, The Netherlands

Funding

  • Yuan, Chen: This research is supported by the European Union Horizon 2020 research and innovation programme under grant agreement No. 74079 (ALGSTRONGCRYPTO).

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.

References

  1. Alexander Barg, Kathryn Haymaker, Everett W. Howe, Gretchen L. Matthews, and Anthony Várilly-Alvarado. Locally recoverable codes from algebraic curves and surfaces. CoRR, abs/1701.05212, 2017. URL: http://arxiv.org/abs/1701.05212.
  2. Tom Bohman and Peter Keevash. The early evolution of the H-free process. Inventiones mathematicae, 181(2):291-336, August 2010. Google Scholar
  3. Belá Bollobás. Modern Graph Theory. Springer, New York, 1998. Google Scholar
  4. Parikshit Gopalan, Cheng Huang, Huseyin Simitci, and Sergey Yekhanin. On the Locality of Codeword Symbols. IEEE Trans. Information Theory, 58(11):6925-6934, 2012. Google Scholar
  5. Venkatesan Guruswami, Christopher Umans, and Salil Vadhan. Unbalanced Expanders and Randomness Extractors from Parvaresh-Vardy Codes. J. ACM, 56(4):20:1-20:34, July 2009. Google Scholar
  6. Venkatesan Guruswami, Chaoping Xing, and Chen Yuan. How Long Can Optimal Locally Repairable Codes Be? In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2018, August 20-22, 2018 - Princeton, NJ, USA, pages 41:1-41:11, 2018. Google Scholar
  7. Shlomo Hoory. The Size of Bipartite Graphs with a Given Girth. J. Comb. Theory, Ser. B, 86(2):215-220, 2002. Google Scholar
  8. Cheng Huang, Minghua Chen, and Jin Li. Pyramid Codes: Flexible Schemes to Trade Space for Access Efficiency in Reliable Data Storage Systems. In Sixth IEEE International Symposium on Network Computing and Applications (NCA 2007), 12 - 14 July 2007, Cambridge, MA, USA, pages 79-86, 2007. Google Scholar
  9. Lingfei Jin. Explicit construction of optimal locally recoverable codes of distance 5 and 6 via binary constant weight codes. CoRR, abs/1808.04558, 2018. URL: http://arxiv.org/abs/1808.04558.
  10. Lingfei Jin, Liming Ma, and Chaoping Xing. Construction of optimal locally repairable codes via automorphism groups of rational function fields. CoRR, abs/1710.09638, 2017. URL: http://arxiv.org/abs/1710.09638.
  11. Xudong Li, Liming Ma, and Chaoping Xing. Optimal locally repairable codes via elliptic curves. CoRR, abs/1712.03744, 2017. URL: http://arxiv.org/abs/1712.03744.
  12. Yuan Luo, Chaoping Xing, and Chen Yuan. Optimal locally repairable codes of distance 3 and 4 via cyclic codes. CoRR, abs/1801.03623, 2018. URL: http://arxiv.org/abs/1801.03623.
  13. N. Prakash, Govinda M. Kamath, V. Lalitha, and P. Vijay Kumar. Optimal linear codes with a local-error-correction property. In Proceedings of the 2012 IEEE International Symposium on Information Theory, ISIT 2012, Cambridge, MA, USA, July 1-6, 2012, pages 2776-2780, 2012. Google Scholar
  14. Natalia Silberstein, Ankit Singh Rawat, Onur Ozan Koyluoglu, and Sriram Vishwanath. Optimal locally repairable codes via rank-metric codes. In Proceedings of the 2013 IEEE International Symposium on Information Theory, Istanbul, Turkey, July 7-12, 2013, pages 1819-1823, 2013. Google Scholar
  15. Itzhak Tamo and Alexander Barg. A Family of Optimal Locally Recoverable Codes. IEEE Trans. Information Theory, 60(8):4661-4676, 2014. Google Scholar
  16. Itzhak Tamo, Dimitris S. Papailiopoulos, and Alexandros G. Dimakis. Optimal Locally Repairable Codes and Connections to Matroid Theory. IEEE Trans. Information Theory, 62(12):6661-6671, 2016. Google Scholar
  17. Craig Timmons and Jacques Verstraëte. A counterexample to sparse removal. European Journal of Combinatorics, 44:77-86, 2015. Google Scholar
  18. Jacques Verstraëte. Personal communication. Google Scholar
  19. Jacques Verstraëte. Extremal problems for cycles in graphs, pages 83-116. Springer International Publishing, Cham, 2016. Google Scholar
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