Document Open Access Logo

When Can an Expander Code Correct Ω(n) Errors in O(n) Time?

Authors Kuan Cheng , Minghui Ouyang , Chong Shangguan , Yuanting Shen

PDF

File

Thumbnail PDF
PDF
LIPIcs.APPROX-RANDOM.2024.61.pdf
  • Filesize: 0.88 MB
  • 23 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Error-correcting codes
  • Mathematics of computing → Coding theory
  • Mathematics of computing → Combinatoric problems
Keywords
  • expander codes
  • expander graphs
  • linear-time decoding

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph G together with a linear inner code C₀. Expander codes are Tanner codes whose defining bipartite graph G has good expansion property. This paper is motivated by the following natural and fundamental problem in decoding expander codes:
What are the sufficient and necessary conditions that δ and d₀ must satisfy, so that every bipartite expander G with vertex expansion ratio δ and every linear inner code C₀ with minimum distance d₀ together define an expander code that corrects Ω(n) errors in O(n) time?
For C₀ being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT'96) showed that δ > 3/4 is sufficient; later Viderman (ACM-TOCT'13) improved this to δ > 2/3-Ω(1) and he also showed that δ > 1/2 is necessary. For general linear code C₀, the previously best-known result of Dowling and Gao (IEEE-TIT'18) showed that d₀ = Ω(cδ^{-2}) is sufficient, where c is the left-degree of G.
In this paper, we give a near-optimal solution to the above question for general C₀ by showing that δ d₀ > 3 is sufficient and δ d₀ > 1 is necessary, thereby also significantly improving Dowling-Gao’s result. We present two novel algorithms for decoding expander codes, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius.

Cite As Get BibTex

Kuan Cheng, Minghui Ouyang, Chong Shangguan, and Yuanting Shen. When Can an Expander Code Correct Ω(n) Errors in O(n) Time?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 61:1-61:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.61

Author Details

Kuan Cheng
  • Center on Frontiers of Computing Studies, School of Computer Science, Peking University, China
Minghui Ouyang
  • School of Mathematical Sciences, Peking University, China
Chong Shangguan
  • Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, China
  • Frontiers Science Center for Nonlinear Expectations, Ministry of Education, Qingdao, China
Yuanting Shen
  • Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, China

Funding

The research of Chong Shangguan is supported by the National Key Research and Development Program of China under Grant No. 2021YFA1001000, the National Natural Science Foundation of China under Grant Nos. 12101364 and 12231014, and the Natural Science Foundation of Shandong Province under Grant No. ZR2021QA005.

Acknowledgements

M. Ouyang would like to thank Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS, Daejeon, South Korea) for hosting his visit at the end of 2023.

References

  1. Noga Alon and Michael Capalbo. Explicit unique-neighbor expanders. In The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings., pages 73-79. IEEE, 2002. Google Scholar
  2. Sanjeev Arora, Constantinos Daskalakis, and David Steurer. Message-passing algorithms and improved lp decoding. IEEE Trans. Inf. Theory, 58(12):7260-7271, 2012. URL: https://doi.org/10.1109/TIT.2012.2208584.
  3. Ron Asherov and Irit Dinur. Bipartite unique neighbour expanders via ramanujan graphs. Entropy, 26(4):348, 2024. Google Scholar
  4. Oren Becker. Symmetric unique neighbor expanders and good ldpc codes. Discrete Applied Mathematics, 211:211-216, 2016. Google Scholar
  5. Avraham Ben-Aroya and Amnon Ta-Shma. A combinatorial construction of almost-ramanujan graphs using the zig-zag product. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 325-334, 2008. Google Scholar
  6. Nikolas P. Breuckmann and Jens Niklas Eberhardt. Balanced product quantum codes. IEEE Trans. Inf. Theory, 67(10):6653-6674, 2021. URL: https://doi.org/10.1109/TIT.2021.3097347.
  7. Nikolas P Breuckmann and Jens Niklas Eberhardt. Quantum low-density parity-check codes. PRX Quantum, 2(4):040101, 2021. Google Scholar
  8. Michael Capalbo, Omer Reingold, Salil Vadhan, and Avi Wigderson. Randomness conductors and constant-degree lossless expanders. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 659-668, 2002. Google Scholar
  9. Xue Chen, Kuan Cheng, Xin Li, and Minghui Ouyang. Improved decoding of expander codes. IEEE Trans. Inf. Theory, 69(6):3574-3589, 2023. URL: https://doi.org/10.1109/TIT.2023.3239163.
  10. Kuan Cheng, Minghui Ouyang, Chong Shangguan, and Yuanting Shen. Improved decoding of expander codes: fundamental trade-off between expansion ratio and minimum distance of inner code. arXiv preprint, 2023. URL: https://arxiv.org/abs/2312.16087.
  11. Shashi Kiran Chilappagari, Dung Viet Nguyen, Bane Vasic, and Michael W. Marcellin. On trapping sets and guaranteed error correction capability of LDPC codes and GLDPC codes. IEEE Trans. Inf. Theory, 56(4):1600-1611, 2010. URL: https://doi.org/10.1109/TIT.2010.2040962.
  12. Sae-Young Chung, G David Forney, Thomas J Richardson, and Rüdiger Urbanke. On the design of low-density parity-check codes within 0.0045 db of the shannon limit. IEEE Communications letters, 5(2):58-60, 2001. Google Scholar
  13. Itay Cohen, Roy Roth, and Amnon Ta-Shma. Hdx condensers. In 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS), pages 1649-1664. IEEE, 2023. Google Scholar
  14. Alexandros G. Dimakis, Roxana Smarandache, and Pascal O. Vontobel. Ldpc codes for compressed sensing. IEEE Trans. Inf. Theory, 58(5):3093-3114, 2012. URL: https://doi.org/10.1109/TIT.2011.2181819.
  15. Irit Dinur, Min-Hsiu Hsieh, Ting-Chun Lin, and Thomas Vidick. Good quantum LDPC codes with linear time decoders. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 905-918. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585101.
  16. Michael Dowling and Shuhong Gao. Fast decoding of expander codes. IEEE Trans. Inf. Theory, 64(2):972-978, 2018. URL: https://doi.org/10.1109/TIT.2017.2726064.
  17. Shai Evra, Tali Kaufman, and Gilles Zémor. Decodable quantum LDPC codes beyond the √n distance barrier using high dimensional expanders. CoRR, abs/2004.07935, 2020. URL: https://arxiv.org/abs/2004.07935.
  18. Jon Feldman, Tal Malkin, Rocco A. Servedio, Cliff Stein, and Martin J. Wainwright. Lp decoding corrects a constant fraction of errors. IEEE Trans. Inf. Theory, 53(1):82-89, 2007. URL: https://doi.org/10.1109/TIT.2006.887523.
  19. Robert Gallager. Low-density parity-check codes. IRE Transactions on information theory, 8(1):21-28, 1962. Google Scholar
  20. Louis Golowich. New explicit constant-degree lossless expanders. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 4963-4971. SIAM, 2024. Google Scholar
  21. Shouzhen Gu, Christopher A. Pattison, and Eugene Tang. An efficient decoder for a linear distance quantum LDPC code. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 919-932. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585169.
  22. Venkatesan Guruswami. Iterative decoding of low-density parity check codes (a survey). arXiv preprint cs/0610022, 2006. Google Scholar
  23. Venkatesan Guruswami and Piotr Indyk. Near-optimal linear-time codes for unique decoding and new list-decodable codes over smaller alphabets. In John H. Reif, editor, Proceedings on 34th Annual ACM Symposium on Theory of Computing, May 19-21, 2002, Montréal, Québec, Canada, pages 812-821. ACM, 2002. URL: https://doi.org/10.1145/509907.510023.
  24. Matthew B. Hastings, Jeongwan Haah, and Ryan O'Donnell. Fiber bundle codes: breaking the n^1/2 polylog(n) barrier for quantum LDPC codes. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1276-1288. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451005.
  25. Jun-Ting Hsieh, Theo McKenzie, Sidhanth Mohanty, and Pedro Paredes. Explicit two-sided unique-neighbor expanders. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 788-799, 2024. Google Scholar
  26. Tali Kaufman and Izhar Oppenheim. Construction of new local spectral high dimensional expanders. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 773-786, 2018. Google Scholar
  27. Tali Kaufman and Ran J. Tessler. New cosystolic expanders from tensors imply explicit quantum LDPC codes with Ω(√n log^k n) distance. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC '21: 53rd Annual ACM SIGACT Symposium on Theory of Computing, Virtual Event, Italy, June 21-25, 2021, pages 1317-1329. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451029.
  28. Swastik Kopparty, Noga Ron-Zewi, and Shubhangi Saraf. Simple constructions of unique neighbor expanders from error-correcting codes. arXiv preprint, 2023. URL: https://arxiv.org/abs/2310.19149.
  29. Anthony Leverrier, Jean-Pierre Tillich, and Gilles Zémor. Quantum expander codes. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 810-824. IEEE Computer Society, 2015. URL: https://doi.org/10.1109/FOCS.2015.55.
  30. Anthony Leverrier and Gilles Zémor. Quantum tanner codes. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 872-883. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00117.
  31. Anthony Leverrier and Gilles Zémor. Decoding quantum tanner codes. IEEE Trans. Inf. Theory, 69(8):5100-5115, 2023. URL: https://doi.org/10.1109/TIT.2023.3267945.
  32. Anthony Leverrier and Gilles Zémor. Efficient decoding up to a constant fraction of the code length for asymptotically good quantum codes. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 1216-1244. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH45.
  33. Ting-Chun Lin and Min-Hsiu Hsieh. Good quantum ldpc codes with linear time decoder from lossless expanders. arXiv preprint, 2022. URL: https://arxiv.org/abs/2203.03581.
  34. Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. Ramanujan graphs. Combinatorica, 8(3):261-277, 1988. Google Scholar
  35. Michael G Luby, Michael Mitzenmacher, Mohammad Amin Shokrollahi, and Daniel A Spielman. Efficient erasure correcting codes. IEEE Trans. Inf. Theory, 47(2):569-584, 2001. Google Scholar
  36. GA Margulis. Explicit constructions of expanders (russian), problemy peredaci informacii 9 (1973), no. 4, 71-80. MR0484767, 1973. Google Scholar
  37. Moshe Morgenstern. Existence and explicit constructions of q+ 1 regular ramanujan graphs for every prime power q. Journal of Combinatorial Theory, Series B, 62(1):44-62, 1994. Google Scholar
  38. Jonathan Mosheiff, Nicolas Resch, Noga Ron-Zewi, Shashwat Silas, and Mary Wootters. Ldpc codes achieve list decoding capacity. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 458-469, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00050.
  39. Pavel Panteleev and Gleb Kalachev. Asymptotically good quantum and locally testable classical LDPC codes. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 375-388. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520017.
  40. Pavel Panteleev and Gleb Kalachev. Quantum LDPC codes with almost linear minimum distance. IEEE Trans. Inf. Theory, 68(1):213-229, 2022. URL: https://doi.org/10.1109/TIT.2021.3119384.
  41. Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 3-13. IEEE, 2000. Google Scholar
  42. Thomas J Richardson, Mohammad Amin Shokrollahi, and Rüdiger L Urbanke. Design of capacity-approaching irregular low-density parity-check codes. IEEE Trans. Inf. Theory, 47(2):619-637, 2001. Google Scholar
  43. Thomas J Richardson and Rüdiger L Urbanke. The capacity of low-density parity-check codes under message-passing decoding. IEEE Trans. Inf. Theory, 47(2):599-618, 2001. Google Scholar
  44. Noga Ron-Zewi, Mary Wootters, and Gilles Zémor. Linear-time erasure list-decoding of expander codes. IEEE Trans. Inf. Theory, 67(9):5827-5839, 2021. URL: https://doi.org/10.1109/TIT.2021.3086805.
  45. Ron M. Roth and Vitaly Skachek. Improved nearly-mds expander codes. IEEE Trans. Inf. Theory, 52(8):3650-3661, 2006. URL: https://doi.org/10.1109/TIT.2006.878232.
  46. Michael Sipser and Daniel A Spielman. Expander codes. IEEE Trans. Inf. Theory, 42(6):1710-1722, 1996. Google Scholar
  47. Vitaly Skachek. Minimum distance bounds for expander codes. In 2008 Information Theory and Applications Workshop, pages 366-370. IEEE, 2008. Google Scholar
  48. DA Spielman. Linear-time encodable and decodable error-correcting codes. IEEE Trans. Inf. Theory, 42(6):1723-1731, 1996. Google Scholar
  49. R Tanner. A recursive approach to low complexity codes. IEEE Trans. Inf. Theory, 27(5):533-547, 1981. Google Scholar
  50. Michael Viderman. Linear-time decoding of regular expander codes. ACM Transactions on Computation Theory (TOCT), 5(3):1-25, 2013. Google Scholar
  51. Michael Viderman. LP decoding of codes with expansion parameter above 2/3. Inf. Process. Lett., 113(7):225-228, 2013. URL: https://doi.org/10.1016/J.IPL.2013.01.012.
  52. Gilles Zémor. On expander codes. IEEE Trans. Inf. Theory, 47(2):835-837, 2001. URL: https://doi.org/10.1109/18.910593.
  53. Gillés Zémor. On expander codes. IEEE Transactions on Information Theory, 47(2):835-837, 2001. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact