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Repairing Schemes for Tamo-Barg Codes (2312.12803v2)

Published 20 Dec 2023 in cs.IT and math.IT

Abstract: In this paper, the repair problem for erasures beyond locality in locally repairable codes is explored under a practical system setting, where a rack-aware storage system consists of racks, each containing a few parity checks. This is referred to as a rack-aware system with locality. Two repair schemes are devised to reduce the repair bandwidth for Tamo-Barg codes under the rack-aware model by setting each repair set as a rack. Additionally, a cut-set bound for locally repairable codes under the rack-aware model with locality is introduced. Using this bound, the second repair scheme is proven to be optimal. Furthermore, the partial-repair problem is considered for locally repairable codes under the rack-aware model with locality, and both repair schemes and bounds are introduced for this scenario.

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References (42)
  1. V. R. Cadambe, S. A. Jafar, H. Maleki, K. Ramchandran, and C. Suh, “Asymptotic interference alignment for optimal repair of mds codes in distributed storage,” IEEE Transactions on Information Theory, vol. 59, no. 5, pp. 2974–2987, 2013.
  2. V. R. Cadambe and A. Mazumdar, “Bounds on the size of locally recoverable codes,” IEEE Transactions on information theory, vol. 61, no. 11, pp. 5787–5794, 2015.
  3. H. Cai, M. Cheng, C. Fan, and X. Tang, “Optimal locally repairable systematic codes based on packings,” IEEE Transactions on Communications, vol. 67, no. 1, pp. 39–49, 2019.
  4. H. Cai, Y. Miao, M. Schwartz, and X. Tang, “On optimal locally repairable codes with multiple disjoint repair sets,” IEEE Transactions on Information Theory, vol. 66, no. 4, pp. 2402–2416, 2020.
  5. ——, “On optimal locally repairable codes with super-linear length,” IEEE Transactions on Information Theory, vol. 66, no. 8, pp. 4853–4868, 2020.
  6. H. Cai and M. Schwartz, “On optimal locally repairable codes and generalized sector-disk codes,” IEEE Transactions on Information Theory, vol. 67, no. 2, pp. 686–704, 2021.
  7. B. Chen, W. Fang, S.-T. Xia, J. Hao, and F.-W. Fu, “Improved bounds and Singleton-optimal constructions of locally repairable codes with minimum distance 5 and 6,” IEEE Transactions on Information Theory, vol. 67, no. 1, pp. 217–231, 2021.
  8. Z. Chen, M. Ye, and A. Barg, “Enabling optimal access and error correction for the repair of Reed-Solomon codes,” arXiv preprint arXiv:2001.07189, 2020.
  9. A. G. Dimakis, P. B. Godfrey, Y. Wu, M. J. Wainwright, and K. Ramchandran, “Network coding for distributed storage systems,” IEEE Transactions on information theory, vol. 56, no. 9, pp. 4539–4551, 2010.
  10. M. Forbes and S. Yekhanin, “On the locality of codeword symbols in non-linear codes,” Discrete mathematics, vol. 324, pp. 78–84, 2014.
  11. P. Gopalan, C. Huang, H. Simitci, and S. Yekhanin, “On the locality of codeword symbols,” IEEE Transactions on Information Theory, vol. 58, no. 11, pp. 6925–6934, 2012.
  12. V. Guruswami and M. Wootters, “Repairing Reed-Solomon codes,” IEEE Transactions on Information Theory, vol. 63, no. 9, pp. 5684–5698, 2017.
  13. J. Hao, S.-T. Xia, K. W. Shum, B. Chen, F.-W. Fu, and Y. Yang, “Bounds and constructions of locally repairable codes: parity-check matrix approach,” IEEE Transactions on Information Theory, vol. 66, no. 12, pp. 7465–7474, 2020.
  14. T. Ho, M. Médard, R. Koetter, D. R. Karger, M. Effros, J. Shi, and B. Leong, “A random linear network coding approach to multicast,” IEEE Transactions on information theory, vol. 52, no. 10, pp. 4413–4430, 2006.
  15. L. Holzbaur, S. Puchinger, E. Yaakobi, and A. Wachter-Zeh, “Partial MDS codes with local regeneration,” arXiv preprint arXiv:2001.04711, 2020.
  16. H. Hou, P. P. Lee, K. W. Shum, and Y. Hu, “Rack-aware regenerating codes for data centers,” IEEE Transactions on Information Theory, vol. 65, no. 8, pp. 4730–4745, 2019.
  17. Y. Hu, P. P. Lee, and X. Zhang, “Double regenerating codes for hierarchical data centers,” in 2016 IEEE International Symposium on Information Theory Proceedings.   IEEE, 2016, pp. 245–249.
  18. C. Huang, M. Chen, and J. Li, “Pyramid codes: Flexible schemes to trade space for access efficiency in reliable data storage systems,” ACM Transactions on Storage (TOS), vol. 9, no. 1, pp. 1–28, 2013.
  19. L. Jin, L. Ma, and C. Xing, “Construction of optimal locally repairable codes via automorphism groups of rational function fields,” IEEE Transactions on Information Theory, vol. 66, no. 1, pp. 210–221, 2020.
  20. G. M. Kamath, N. Prakash, V. Lalitha, and P. V. Kumar, “Codes with local regeneration and erasure correction,” IEEE Transactions on Information Theory, vol. 60, no. 8, pp. 4637–4660, 2014.
  21. G. Kim and J. Lee, “Locally repairable codes with unequal local erasure correction,” IEEE Transactions on Information Theory, vol. 64, no. 11, pp. 7137–7152, 2018.
  22. J. Li and X. Tang, “Optimal exact repair strategy for the parity nodes of the (k+2,k)𝑘2𝑘(k+2,k)( italic_k + 2 , italic_k ) zigzag code,” IEEE Transactions on Information Theory, vol. 62, no. 9, pp. 4848–4856, 2016.
  23. J. Li, X. Tang, H. Hou, Y. S. Han, B. Bai, and G. Zhang, “Pmds array codes with small sub-packetization, small repair bandwidth/rebuilding access,” IEEE Transactions on Information Theory, vol. 69, no. 3, pp. 1551–1566, 2022.
  24. J. Li, X. Tang, and U. Parampalli, “A framework of constructions of minimal storage regenerating codes with the optimal access/update property,” IEEE Transactions on Information theory, vol. 61, no. 4, pp. 1920–1932, 2015.
  25. J. Li, X. Tang, and C. Tian, “A generic transformation to enable optimal repair in mds codes for distributed storage systems,” IEEE Transactions on Information Theory, vol. 64, no. 9, pp. 6257–6267, 2018.
  26. X. Li, L. Ma, and C. Xing, “Construction of asymptotically good locally repairable codes via automorphism groups of function fields,” IEEE Transactions on Information Theory, vol. 65, no. 11, pp. 7087–7094, 2019.
  27. J. Liu, S. Mesnager, and L. Chen, “New constructions of optimal locally recoverable codes via good polynomials,” IEEE Transactions on Information Theory, vol. 64, no. 2, pp. 889–899, 2017.
  28. Y. Liu, J. Li, and X. Tang, “A generic transformation to enable optimal repair/access mds array codes with multiple repair degrees,” IEEE Transactions on Information Theory, 2023.
  29. A. Mazumdar, V. Chandar, and G. W. Wornell, “Update-efficiency and local repairability limits for capacity approaching codes,” IEEE Journal on Selected Areas in Communications, vol. 32, no. 5, pp. 976–988, 2014.
  30. N. Prakash, G. M. Kamath, V. Lalitha, and P. V. Kumar, “Optimal linear codes with a local-error-correction property,” in 2012 IEEE International Symposium on Information Theory Proceedings.   IEEE, 2012, pp. 2776–2780.
  31. A. S. Rawat, O. O. Koyluoglu, N. Silberstein, and S. Vishwanath, “Optimal locally repairable and secure codes for distributed storage systems,” IEEE Transactions on Information Theory, vol. 60, no. 1, pp. 212–236, 2014.
  32. A. S. Rawat, D. S. Papailiopoulos, A. G. Dimakis, and S. Vishwanath, “Locality and availability in distributed storage,” IEEE Transactions on Information Theory, vol. 62, no. 8, pp. 4481–4493, 2016.
  33. B. Sasidharan, G. K. Agarwal, and P. V. Kumar, “Codes with hierarchical locality,” in 2015 IEEE International Symposium on Information Theory Proceedings.   IEEE, 2015, pp. 1257–1261.
  34. I. Tamo and A. Barg, “A family of optimal locally recoverable codes,” IEEE Transactions on Information Theory, vol. 60, no. 8, pp. 4661–4676, 2014.
  35. I. Tamo, A. Barg, and A. Frolov, “Bounds on the parameters of locally recoverable codes,” IEEE Transactions on Information Theory, vol. 62, no. 6, pp. 3070–3083, 2016.
  36. I. Tamo, Z. Wang, and J. Bruck, “Zigzag codes: MDS array codes with optimal rebuilding,” IEEE Transactions on Information Theory, vol. 59, no. 3, pp. 1597–1616, 2012.
  37. I. Tamo, M. Ye, and A. Barg, “Optimal repair of Reed-Solomon codes: Achieving the cut-set bound,” in 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).   IEEE, 2017, pp. 216–227.
  38. ——, “The repair problem for Reed-Solomon codes: Optimal repair of single and multiple erasures with almost optimal node size,” IEEE Transactions on Information Theory, vol. 65, no. 5, pp. 2673–2695, 2018.
  39. A. Wang and Z. Zhang, “Repair locality with multiple erasure tolerance,” IEEE Transactions on Information Theory, vol. 60, no. 11, pp. 6979–6987, 2014.
  40. C. Xing and C. Yuan, “Construction of optimal locally recoverable codes and connection with hypergraph,” in 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019).   Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019.
  41. M. Ye and A. Barg, “Explicit constructions of high-rate MDS array codes with optimal repair bandwidth,” IEEE Transactions on Information Theory, vol. 63, no. 4, pp. 2001–2014, 2017.
  42. A. Zeh and E. Yaakobi, “Bounds and constructions of codes with multiple localities,” in 2016 IEEE International Symposium on Information Theory Proceedings.   IEEE, 2016, pp. 640–644.
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