How Long Can Optimal Locally Repairable Codes Be?

Authors Venkatesan Guruswami , Chaoping Xing , Chen Yuan

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Author Details

Venkatesan Guruswami
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, USA.
Chaoping Xing
  • School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore.
Chen Yuan
  • Centrum Wiskunde & Informatica, Amsterdam, Netherlands.

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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). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 41:1-41:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


A locally repairable code (LRC) with locality r allows for the recovery of any erased codeword symbol using only r other codeword symbols. A Singleton-type bound dictates the best possible trade-off between the dimension and distance of LRCs - an LRC attaining this trade-off is deemed optimal. Such optimal LRCs have been constructed over alphabets growing linearly in the block length. Unlike the classical Singleton bound, however, it was not known if such a linear growth in the alphabet size is necessary, or for that matter even if the alphabet needs to grow at all with the block length. Indeed, for small code distances 3,4, arbitrarily long optimal LRCs were known over fixed alphabets. Here, we prove that for distances d >=slant 5, the code length n of an optimal LRC over an alphabet of size q must be at most roughly O(d q^3). For the case d=5, our upper bound is O(q^2). We complement these bounds by showing the existence of optimal LRCs of length Omega_{d,r}(q^{1+1/floor[(d-3)/2]}) when d <=slant r+2. Our bounds match when d=5, pinning down n=Theta(q^2) as the asymptotically largest length of an optimal LRC for this case.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Locally Repairable Code
  • Singleton Bound


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  1. S. Ball. On large subsets of a finite vector space in which every subset of basis size is a basis. J. Eur, 14:733-748, October 2012. Google Scholar
  2. A. Barg, K. Haymaker, E. Howe, G. Matthews, and A. V́arilly-Alvarado. Locally recoverable codes from algebraic curves and surfaces. In E. W. Howe, K. E. Lauter, and J. L. Walker, editors, Algebraic Geometry for Coding Theory and Cryptography, pages 95-126. s, Springer, 2017. Google Scholar
  3. A. Barg, I. Tamo, and S. Vlăduţ. Locally recoverable codes on algebraic curves. IEEE Trans. Inform.Theory, 63:4928-4939, 2017. Google Scholar
  4. V. Cadambe and A. Mazumda. Bounds on the size of locally recoverable codes. IEEE Trans. Inform.Theory, 61:5787-5794, 2015. Google Scholar
  5. M. Forbes and S. Yekhanin. On the locality of codeword symbols in non-linear codes. Discrete Mathematics, 324(6):78-84, 2014. Google Scholar
  6. P. Gopalan, C. Huang, H. Simitci, and S. Yekhanin. On the locality of codeword symbols. IEEE Trans. Inform.Theory, 58:6925-6934, 2012. Google Scholar
  7. S. Gopi, V. Guruswami, and S. Yekhanin. On maximally recoverable local reconstruction codes. Electronic Colloquium on Computational Complexity, 24::183, 2017. Google Scholar
  8. J. Han and L. A. Lastras-Montano. Reliable memories with subline accesses. In Proc. IEEE Internat. Sympos. Inform. Theory, pages 2531-2535, 2007. Google Scholar
  9. C. Huang, M. Chen, and J. 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, pages 79-86, 2007. Google Scholar
  10. C. Huang, H. Simitci, Y. Xu, Ogus A., B. Calder, P. Gopalan, J. Li, and S. Yekhanin. Erasure coding in windows azure storage. In USENIX Annual Technical Conference (ATC), pages 15-26, 2012. Google Scholar
  11. L. Jin, L. Ma, and Xing C. Construction of optimal locally repairable codes via automorphism groups of rational function fields. URL:
  12. O. Kolosov, A. Barg, I. Tamo, and G. Yadgar. Optimal lrc codes for all lengths n ⩽ q. URL:
  13. X. Li, L. Ma, and C. Xing. Optimal locally repairable codes via elliptic curves. To appear in IEEE Trans. Inf. Theory, 2017. URL:
  14. Y. Luo, C. Xing, and C. Yuan. Optimal locally repairable codes of distance 3 and 4 via cyclic codes. To appear in IEEE Trans. Inf. Theory, 2018. URL:
  15. D. S. Papailiopoulos and A. G. Dimakis. Locally repairable codes. IEEE Trans. Inform.Theory, 60:5843-5855, 2014. Google Scholar
  16. N. Prakash, G. M. Kamath, V. Lalitha, and P. V. Kumar. Optimal linear codes with a local-error-correction property. In Proc. 2012 IEEE Int. Symp. Inform. Theory, pages 2776-2780, 2012. Google Scholar
  17. M. Sathiamoorthy, M. Asteris, D. S. Papailiopoulos, A. G. Dimakis, R. Vadali, S. Chen, and D. Borthakur. XORing elephants: novel erasure codes for big data. Proceedings of VLDB Endowment (PVLDB), pages 325-336, 2013. Google Scholar
  18. N. Silberstein, A. S. Rawat, O. O. Koyluoglu, and S. Vichwanath. Optimal locally repairable codes via rank-matric codes. In Proc. IEEE Int. Symp. Inf. Theory, pages 1819-1823, 2013. Google Scholar
  19. I. Tamo and A. Barg. A family of optimal locally recoverable codes. IEEE Trans. Inform.Theory, 60:4661-4676, 2014. Google Scholar
  20. I. Tamo, D. S. Papailiopoulos, and A. G. Dimakis. Optimal locally repairable codes and connections to matroid theory. IEEE Trans. Inform.Theory, 62:6661-6671, 2016. Google Scholar
  21. Z. Zhang, J. Xu, and M. Liu. Constructions of optimal locally repairable codes over small fields. SCIENTIA SINICA Mathematica, 47(11):1607-1614, 2017. Google Scholar