Lifted Multiplicity Codes and the Disjoint Repair Group Property

Authors Ray Li, Mary Wootters

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Ray Li
  • Department of Computer Science, Stanford University, CA, USA
Mary Wootters
  • Departments of Computer Science and Electrical Engineering, Stanford University, CA, USA


We thank Eitan Yaakobi for helpful conversations. We thank Julien Lavauzelle for pointing out the reference [Wu, 2015] and also for pointing out an error in the proof of the original version of Proposition 18. We thank anonymous reviewers for helpful comments on an earlier draft of this paper.

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Ray Li and Mary Wootters. Lifted Multiplicity Codes and the Disjoint Repair Group Property. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 38:1-38:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Lifted Reed Solomon Codes (Guo, Kopparty, Sudan 2013) were introduced in the context of locally correctable and testable codes. They are multivariate polynomials whose restriction to any line is a codeword of a Reed-Solomon code. We consider a generalization of their construction, which we call lifted multiplicity codes. These are multivariate polynomial codes whose restriction to any line is a codeword of a multiplicity code (Kopparty, Saraf, Yekhanin 2014). We show that lifted multiplicity codes have a better trade-off between redundancy and a notion of locality called the t-disjoint-repair-group property than previously known constructions. More precisely, we show that, for t <=sqrt{N}, lifted multiplicity codes with length N and redundancy O(t^{0.585} sqrt{N}) have the property that any symbol of a codeword can be reconstructed in t different ways, each using a disjoint subset of the other coordinates. This gives the best known trade-off for this problem for any super-constant t < sqrt{N}. We also give an alternative analysis of lifted Reed Solomon codes using dual codes, which may be of independent interest.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Lifted codes
  • Multiplicity codes
  • Disjoint repair group property
  • PIR code
  • Coding theory


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  1. Hilal Asi and Eitan Yaakobi. Nearly optimal constructions of PIR and batch codes. IEEE Transactions on Information Theory, 65(2):947-964, 2019. Google Scholar
  2. Simon R. Blackburn and Tuvi Etzion. PIR Array Codes with Optimal PIR Rate. ArXiv e-prints, July 2016. URL:
  3. Francis Y Chin. A generalized asymptotic upper bound on fast polynomial evaluation and interpolation. SIAM Journal on Computing, 5(4):682-690, 1976. Google Scholar
  4. Arman Fazeli, Alexander Vardy, and Eitan Yaakobi. Codes for distributed PIR with low storage overhead. In 2015 IEEE International Symposium on Information Theory (ISIT), pages 2852-2856. IEEE, 2015. Google Scholar
  5. S. Luna Frank-Fischer, Venkatesan Guruswami, and Mary Wootters. Locality via Partially Lifted Codes. CoRR, abs/1704.08627, 2017. URL:
  6. Alan Guo, Swastik Kopparty, and Madhu Sudan. New affine-invariant codes from lifting. In Innovations in Theoretical Computer Science, ITCS '13, Berkeley, CA, USA, January 9-12, 2013, pages 529-540, 2013. URL:
  7. Venkatesan Guruswami and Carol Wang. Linear-algebraic list decoding for variants of Reed-Solomon codes. IEEE Transactions on Information Theory, 59(6):3257-3268, 2013. Google Scholar
  8. Brett Hemenway, Rafail Ostrovsky, and Mary Wootters. Local correctability of expander codes. Information and Computation, 243:178-190, 2015. Google Scholar
  9. James W. P. Hirschfeld, Gábor Korchmáros, and Fernando Torres. Algebraic Curves over a Finite Field. Princeton Series in Applied Mathematics. Princeton University Press, 2008. Google Scholar
  10. Yuval Ishai, Eyal Kushilevitz, Rafail Ostrovsky, and Amit Sahai. Batch codes and their applications. In Proceedings of the thirty-sixth annual ACM Symposium on the Theory of Computing, STOC 2004, pages 262-271. ACM, 2004. Google Scholar
  11. Jonathan Katz and Luca Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In Proceedings of the 32nd symposium on Theory of Computing, STOC 2000, pages 80-86, 2000. URL:
  12. Swastik Kopparty. List-decoding multiplicity codes. Theory of Computing, 11(1):149-182, 2015. Google Scholar
  13. Swastik Kopparty. Some remarks on multiplicity codes. CoRR, abs/1505.07547, 2015. URL:
  14. Swastik Kopparty, Or Meir, Noga Ron-Zewi, and Shubhangi Saraf. High-rate locally-correctable and locally-testable codes with sub-polynomial query complexity. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, pages 202-215. ACM, 2016. Google Scholar
  15. Swastik Kopparty, Shubhangi Saraf, and Sergey Yekhanin. High-rate codes with sublinear-time decoding. Journal of the ACM (JACM), 61(5):28, 2014. Google Scholar
  16. Shu Lin and Daniel J Costello. Error control coding. Pearson Education India, 2001. Google Scholar
  17. Sankeerth Rao and Alexander Vardy. Lower Bound on the Redundancy of PIR Codes. arXiv preprint arXiv:1605.01869, 2016. URL:
  18. Ankit Singh Rawat, Dimitris S Papailiopoulos, Alexandros G Dimakis, and Sriram Vishwanath. Locality and availability in distributed storage. In 2014 IEEE International Symposium on Information Theory, pages 681-685. IEEE, 2014. Google Scholar
  19. Ankit Singh Rawat, Zhao Song, Alexandros G Dimakis, and Anna Gál. Batch codes through dense graphs without short cycles. IEEE Transactions on Information Theory, 62(4):1592-1604, 2016. Google Scholar
  20. Vitaly Skachek. Batch and PIR codes and their connections to locally repairable codes. In Network Coding and Subspace Designs, pages 427-442. Springer, 2018. Google Scholar
  21. Itzhak Tamo and Alexander Barg. Bounds on locally recoverable codes with multiple recovering sets. In 2014 IEEE International Symposium on Information Theory, pages 691-695. IEEE, 2014. Google Scholar
  22. Itzhak Tamo, Alexander Barg, and Alexey Frolov. Bounds on the parameters of locally recoverable codes. IEEE Transactions on Information Theory, 62(6):3070-3083, 2016. Google Scholar
  23. Anyu Wang and Zhifang Zhang. Repair locality with multiple erasure tolerance. IEEE Transactions on Information Theory, 60(11):6979-6987, 2014. Google Scholar
  24. David P. Woodruff. A Quadratic Lower Bound for Three-Query Linear Locally Decodable Codes over Any Field, pages 766-779. Springer Berlin Heidelberg, Berlin, Heidelberg, 2010. URL:
  25. Mary Wootters. Linear codes with disjoint repair groups. Not intended for publication, available at, 2016.
  26. Liyasi Wu. Revisiting the multiplicity codes: A new class of high-rate locally correctable codes. In 2015 53rd Annual Allerton Conference on Communication, Control, and Computing (Allerton), pages 509-513. IEEE, 2015. Google Scholar
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