Document Open Access Logo

New Near-Linear Time Decodable Codes Closer to the GV Bound

Authors Guy Blanc, Dean Doron

PDF
Thumbnail PDF

File

LIPIcs.CCC.2022.10.pdf
  • Filesize: 1.05 MB
  • 40 pages

Document Identifiers

Author Details

Guy Blanc
  • Computer Science Department, Stanford University, CA, USA
Dean Doron
  • Department of Computer Science, Ben Gurion University of the Negev, Beer Sheva, Israel

Funding

  • Blanc, Guy: Supported by NSF CAREER Award 1942123.
  • Doron, Dean: Part of this work was done while at Stanford, supported by NSF Award CCF-1763311.

Acknowledgements

We are grateful to Victor Lecomte and Omer Reingold for stimulating discussions and collaboration in the early stages of the project. We also thank Ori Parzanchevski, Madhur Tulsiani, and Mary Wootters and for interesting and useful discussions.

Cite As Get BibTex

Guy Blanc and Dean Doron. New Near-Linear Time Decodable Codes Closer to the GV Bound. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 10:1-10:40, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CCC.2022.10

Abstract

We construct a family of binary codes of relative distance 1/2-ε and rate ε² ⋅ 2^(-log^α (1/ε)) for α ≈ 1/2 that are decodable, probabilistically, in near-linear time. This improves upon the rate of the state-of-the-art near-linear time decoding near the GV bound due to Jeronimo, Srivastava, and Tulsiani, who gave a randomized decoding of Ta-Shma codes with α ≈ 5/6 [Ta-Shma, 2017; Jeronimo et al., 2021]. Each code in our family can be constructed in probabilistic polynomial time, or deterministic polynomial time given sufficiently good explicit 3-uniform hypergraphs. 
Our construction is based on a new graph-based bias amplification method. While previous works start with some base code of relative distance 1/2-ε₀ for ε₀ ≫ ε and amplify the distance to 1/2-ε by walking on an expander, or on a carefully tailored product of expanders, we walk over very sparse, highly mixing, hypergraphs. Study of such hypergraphs further offers an avenue toward achieving rate Ω̃(ε²). For our unique- and list-decoding algorithms, we employ the framework developed in [Jeronimo et al., 2021].

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Unique decoding
  • list decoding
  • the Gilbert-Varshamov bound
  • small-bias sample spaces
  • hypergraphs
  • expander walks

Metrics

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

References

  1. Noga Alon. Explicit expanders of every degree and size. Combinatorica, pages 1-17, 2021. Google Scholar
  2. Noga Alon, Jehoshua Bruck, Joseph Naor, Moni Naor, and Ron M. Roth. Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs. Information Theory, IEEE Transactions on, 38(2):509-516, 1992. Google Scholar
  3. Noga Alon, Oded Goldreich, Johan Håstad, and René Peralta. Simple constructions of almost k-wise independent random variables. Random Structures & Algorithms, 3(3):289-304, 1992. Google Scholar
  4. Avraham Ben-Aroya and Amnon Ta-Shma. A combinatorial construction of almost-Ramanujan graphs using the zig-zag product. SIAM Journal on Computing, 40(2):267-290, 2011. Google Scholar
  5. Avraham Ben-Aroya and Amnon Ta-Shma. Constructing small-bias sets from algebraic-geometric codes. Theory of Computing, 9(5):253-272, 2013. Google Scholar
  6. Yonatan Bilu and Shlomo Hoory. On codes from hypergraphs. European Journal of Combinatorics, 25(3):339-354, 2004. Google Scholar
  7. Yonatan Bilu and Nathan Linial. Lifts, discrepancy and nearly optimal spectral gap. Combinatorica, 26(5):495-519, 2006. Google Scholar
  8. Andrej Bogdanov. A different way to improve the bias via expanders. Topics in (and out) the theory of computing, Lecture, 2012. Google Scholar
  9. Emma Cohen, Dhruv Mubayi, Peter Ralli, and Prasad Tetali. Inverse expander mixing for hypergraphs. The Electronic Journal of Combinatorics, 23(2):P2-20, 2016. Google Scholar
  10. David Conlon, Jonathan Tidor, and Yufei Zhao. Hypergraph expanders of all uniformities from Cayley graphs. Proceedings of the London Mathematical Society, 121(5):1311-1336, 2020. Google Scholar
  11. Joel Friedman and Avi Wigderson. On the second eigenvalue of hypergraphs. Combinatorica, 15(1):43-65, 1995. Google Scholar
  12. Edgar N. Gilbert. A comparison of signalling alphabets. The Bell system technical journal, 31(3):504-522, 1952. Google Scholar
  13. Oded Goldreich. On constructing expanders for any number of vertices, October 2019. Available at URL: https://www.wisdom.weizmann.ac.il/~oded/R1/ex4all.pdf.
  14. Konstantin Golubev and Ori Parzanchevski. Spectrum and combinatorics of two-dimensional Ramanujan complexes. Israel Journal of Mathematics, 230(2):583-612, 2019. Google Scholar
  15. Venkatesan Guruswami and Piotr Indyk. Linear-time encodable/decodable codes with near-optimal rate. IEEE Transactions on Information Theory, 51(10):3393-3400, 2005. Google Scholar
  16. Venkatesan Guruswami, Atri Rudra, and Madhu Sudan. Essential Coding Theory, 2015. URL: http://www.cse.buffalo.edu/faculty/atri/courses/coding-theory/book.
  17. Brett Hemenway, Noga Ron-Zewi, and Mary Wootters. Local list recovery of high-rate tensor codes and applications. SIAM Journal on Computing, pages FOCS17-157, 2019. Google Scholar
  18. Wassily Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American statistical association, 58(301):13-30, 1963. Google Scholar
  19. Fernando Granha Jeronimo, Dylan Quintana, Shashank Srivastava, and Madhur Tulsiani. Unique decoding of explicit ε-balanced codes near the Gilbert-Varshamov bound. In Proceedings of the 61st Annual Symposium on Foundations of Computer Science (FOCS 2020), pages 434-445. IEEE, 2020. Google Scholar
  20. Fernando Granha Jeronimo, Shashank Srivastava, and Madhur Tulsiani. Near-linear time decoding of Ta-Shma’s codes via splittable regularity. In Proceedings of the 53rdth Annual Symposium on Theory of Computing (STOC 2021), pages 1527-1536. ACM, 2021. Google Scholar
  21. Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, and Shashwat Silas. On list recovery of high-rate tensor codes. IEEE Transactions on Information Theory, 67(1):296-316, 2020. Google Scholar
  22. Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, and Mary Wootters. Improved decoding of folded Reed-Solomon and multiplicity codes. In Proceedings of the 59th Annual Symposium on Foundations of Computer Science (FOCS 2018), pages 212-223. IEEE, 2018. Google Scholar
  23. John Lenz and Dhruv Mubayi. Eigenvalues and linear quasirandom hypergraphs. In Forum of Mathematics, Sigma, volume 3. Cambridge University Press, 2015. Google Scholar
  24. Ray Li and Mary Wootters. Improved list-decodability of random linear binary codes. In APPROX-RANDOM, volume 116 of LIPIcs, pages 50:1-50:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  25. Eyal Lubetzky and Yuval Peres. Cutoff on all Ramanujan graphs. Geometric and Functional Analysis, 26(4):1190-1216, 2016. Google Scholar
  26. Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. Ramanujan graphs. Combinatorica, 8(3):261-277, 1988. Google Scholar
  27. Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Explicit constructions of Ramanujan complexes of type A_d. European Journal of Combinatorics, 26(6):965-993, 2005. Google Scholar
  28. Grigorii Aleksandrovich Margulis. Explicit group-theoretical constructions of combinatorial schemes and their application to the design of expanders and concentrators. Problemy peredachi informatsii, 24(1):51-60, 1988. Google Scholar
  29. Jack Murtagh, Omer Reingold, Aaron Sidford, and Salil Vadhan. Deterministic approximation of random walks in small space. Theory of Computing, 17(1):1-35, 2021. Google Scholar
  30. Joseph Naor and Moni Naor. Small-bias probability spaces: Efficient constructions and applications. SIAM Journal on Computing, 22(4):838-856, 1993. Google Scholar
  31. Ori Parzanchevski. Mixing in high-dimensional expanders. Combinatorics, Probability and Computing, 26(5):746-761, 2017. Google Scholar
  32. Ori Parzanchevski, Ron Rosenthal, and Ran J. Tessler. Isoperimetric inequalities in simplicial complexes. Combinatorica, 36(2):195-227, 2016. Google Scholar
  33. Michael Sipser and Daniel A. Spielman. Expander codes. IEEE transactions on Information Theory, 42(6):1710-1722, 1996. Google Scholar
  34. Amnon Ta-Shma. Explicit, almost optimal, ε-balanced codes. In Proceedings of the 49th Annual Symposium on Theory of Computing (STOC 2017), pages 238-251. ACM, 2017. Google Scholar
  35. R. Tanner. A recursive approach to low complexity codes. IEEE Transactions on information theory, 27(5):533-547, 1981. Google Scholar
  36. Rom Rubenovich Varshamov. Estimate of the number of signals in error correcting codes. Docklady Akad. Nauk, SSSR, 117:739-741, 1957. Google Scholar
  37. 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
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact