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Visible Rank and Codes with Locality

Authors Omar Alrabiah, Venkatesan Guruswami



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Omar Alrabiah
  • Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, USA
Venkatesan Guruswami
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA

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Omar Alrabiah and Venkatesan Guruswami. Visible Rank and Codes with Locality. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 57:1-57:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.57

Abstract

We propose a framework to study the effect of local recovery requirements of codeword symbols on the dimension of linear codes, based on a combinatorial proxy that we call visible rank. The locality constraints of a linear code are stipulated by a matrix H of ⋆’s and 0’s (which we call a "stencil"), whose rows correspond to the local parity checks (with the ⋆’s indicating the support of the check). The visible rank of H is the largest r for which there is a r × r submatrix in H with a unique generalized diagonal of ⋆’s. The visible rank yields a field-independent combinatorial lower bound on the rank of H and thus the co-dimension of the code. We point out connections of the visible rank to other notions in the literature such as unique restricted graph matchings, matroids, spanoids, and min-rank. In particular, we prove a rank-nullity type theorem relating visible rank to the rank of an associated construct called symmetric spanoid, which was introduced by Dvir, Gopi, Gu, and Wigderson [Zeev Dvir et al., 2020]. Using this connection and a construction of appropriate stencils, we answer a question posed in [Zeev Dvir et al., 2020] and demonstrate that symmetric spanoid rank cannot improve the currently best known Õ(n^{(q-2)/(q-1)}) upper bound on the dimension of q-query locally correctable codes (LCCs) of length n. This also pins down the efficacy of visible rank as a proxy for the dimension of LCCs. We also study the t-Disjoint Repair Group Property (t-DRGP) of codes where each codeword symbol must belong to t disjoint check equations. It is known that linear codes with 2-DRGP must have co-dimension Ω(√n) (which is matched by a simple product code construction). We show that there are stencils corresponding to 2-DRGP with visible rank as small as O(log n). However, we show the second tensor of any 2-DRGP stencil has visible rank Ω(n), thus recovering the Ω(√n) lower bound for 2-DRGP. For q-LCC, however, the k'th tensor power for k ⩽ n^{o(1)} is unable to improve the Õ(n^{(q-2)/(q-1)}) upper bound on the dimension of q-LCCs by a polynomial factor.Inspired by this and as a notion of intrinsic interest, we define the notion of visible capacity of a stencil as the limiting visible rank of high tensor powers, analogous to Shannon capacity, and pose the question whether there can be large gaps between visible capacity and algebraic rank.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Visible Rank
  • Stencils
  • Locality
  • DRGP Codes
  • Locally Correctable Codes

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References

  1. Noga Alon and Eyal Lubetzky. The shannon capacity of a graph and the independence numbers of its powers. IEEE Transactions on Information Theory, 52(5):2172-2176, 2006. Google Scholar
  2. Ziv Bar-Yossef, Yitzhak Birk, T. S. Jayram, and Tomer Kol. Index coding with side information. IEEE Trans. Inf. Theory, 57(3):1479-1494, 2011. Google Scholar
  3. Boaz Barak, Zeev Dvir, Amir Yehudayoff, and Avi Wigderson. Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 519-528, 2011. Google Scholar
  4. Zeev Dvir. Incidence theorems and their applications. arXiv preprint arXiv:1208.5073, 2012. Google Scholar
  5. Zeev Dvir, Sivakanth Gopi, Yuzhou Gu, and Avi Wigderson. Spanoids - an abstraction of spanning structures, and a barrier for LCCs. SIAM J. Comput., 49(3):465-496, 2020. Google Scholar
  6. Arman Fazeli, Alexander Vardy, and Eitan Yaakobi. Codes for distributed PIR with low storage overhead. In IEEE International Symposium on Information Theory, pages 2852-2856, 2015. Google Scholar
  7. Mathew C Francis, Dalu Jacob, and Satyabrata Jana. Uniquely restricted matchings in interval graphs. SIAM Journal on Discrete Mathematics, 32(1):148-172, 2018. Google Scholar
  8. S. Luna Frank-Fischer, Venkatesan Guruswami, and Mary Wootters. Locality via partially lifted codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM, volume 81 of LIPIcs, pages 43:1-43:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  9. Harold N Gabow, Haim Kaplan, and Robert E Tarjan. Unique maximum matching algorithms. Journal of Algorithms, 40(2):159-183, 2001. Google Scholar
  10. Alexander Golovnev, Oded Regev, and Omri Weinstein. The minrank of random graphs. IEEE Trans. Inf. Theory, 64(11):6990-6995, 2018. Google Scholar
  11. Martin Charles Golumbic, Tirza Hirst, and Moshe Lewenstein. Uniquely restricted matchings. Algorithmica, 31(2):139-154, 2001. Google Scholar
  12. Parikshit Gopalan, Cheng Huang, Huseyin Simitci, and Sergey Yekhanin. On the locality of codeword symbols. IEEE Transactions on Information theory, 58(11):6925-6934, 2012. Google Scholar
  13. Sivakanth Gopi. Locality in coding theory. PhD thesis, Princeton University, 2018. Google Scholar
  14. Alan Guo, Swastik Kopparty, and Madhu Sudan. New affine-invariant codes from lifting. In Proceedings of the Innovations in Theoretical Computer Science Conference, pages 529-540, 2013. Google Scholar
  15. Willem H. Haemers. On some problems of Lovász concerning the Shannon capacity of a graph. IEEE Trans. Inf. Theory, 25(2):231-232, 1979. Google Scholar
  16. Thanh Minh Hoang, Meena Mahajan, and Thomas Thierauf. On the bipartite unique perfect matching problem. In International Colloquium on Automata, Languages, and Programming, pages 453-464. Springer, 2006. Google Scholar
  17. Cheng Huang, Huseyin Simitci, Yikang Xu, Aaron Ogus, Brad Calder, Parikshit Gopalan, Jin Li, and Sergey Yekhanin. Erasure coding in windows azure storage. In USENIX Annual Technical Conference, 2012. Google Scholar
  18. Eran Iceland and Alex Samorodnitsky. On coset leader graphs of structured linear codes. Discrete and Computational Geometry, 63:560–576, 2020. Google Scholar
  19. Jonathan Katz and Luca Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pages 80-86, 2000. Google Scholar
  20. Ray Li and Mary Wootters. Lifted multiplicity codes and the disjoint repair group property. IEEE Trans. Inf. Theory, 67(2):716-725, 2021. Google Scholar
  21. László Lovász. Communication complexity: A survey. Technical report, Princeton University TR-204-89, February 1989. University of Zurich, Department of Informatics. Google Scholar
  22. Sounaka Mishra. On the maximum uniquely restricted matching for bipartite graphs. Electronic Notes in Discrete Mathematics, 37:345-350, 2011. Google Scholar
  23. C. Ramya. Recent progress on matrix rigidity-a survey. arXiv preprint arXiv:2009.09460, 2020. Google Scholar
  24. Sankeerth Rao and Alexander Vardy. Lower bound on the redundancy of PIR codes. arXiv preprint arXiv:1605.01869, 2016. Google Scholar
  25. Maheswaran Sathiamoorthy, Megasthenis Asteris, Dimitris Papailiopoulos, Alexandros Dimakis, Ramkumar Vadali, Scott Chen, and Dhruba Borthakur. XORing elephants: Novel erasure codes for big data. Proc. VLDB Endow., 6:325-336, 2013. Google Scholar
  26. Maguy Tréfois. Topics in combinatorial matrix theory. PhD thesis, PhD thesis, UCL, 2016. Google Scholar
  27. Maguy Trefois and Jean-Charles Delvenne. Zero forcing sets, constrained matchings and minimum rank. Linear and Multilinear Algebra, 2013. Google Scholar
  28. Leslie G Valiant. Graph-theoretic arguments in low-level complexity. In International Symposium on Mathematical Foundations of Computer Science, pages 162-176. Springer, 1977. Google Scholar
  29. David P. Woodruff. A quadratic lower bound for three-query linear locally decodable codes over any field. J. Comput. Sci. Technol., 27(4):678-686, 2012. Google Scholar
  30. Mary Wootters. Linear codes with disjoint repair groups. unpublished manuscript, 2016. Google Scholar
  31. Sergey Yekhanin. Locally decodable codes. Foundations and Trends in Theoretical Computer Science, 6(3):139-255, 2012. URL: https://doi.org/10.1561/0400000030.
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