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Near-Optimal Erasure List-Decodable Codes

Authors Avraham Ben-Aroya, Dean Doron, Amnon Ta-Shma

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Avraham Ben-Aroya
  • The Blavatnik School of Computer Science, Tel-Aviv University, Israel
Dean Doron
  • Department of Computer Science, Stanford University, CA, US
Amnon Ta-Shma
  • The Blavatnik School of Computer Science, Tel-Aviv University, Israel

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Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma. Near-Optimal Erasure List-Decodable Codes. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 1:1-1:27, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


A code 𝒞 ⊆ {0,1}^n̅ is (s,L) erasure list-decodable if for every word w, after erasing any s symbols of w, the remaining n̅-s symbols have at most L possible completions into a codeword of 𝒞. Non-explicitly, there exist binary ((1-τ)n̅,L) erasure list-decodable codes with rate approaching τ and tiny list-size L = O(log 1/(τ)). Achieving either of these parameters explicitly is a natural open problem (see, e.g., [Guruswami and Indyk, 2002; Guruswami, 2003; Guruswami, 2004]). While partial progress on the problem has been achieved, no prior nontrivial explicit construction achieved rate better than Ω(τ²) or list-size smaller than Ω(1/τ). Furthermore, Guruswami showed no linear code can have list-size smaller than Ω(1/τ) [Guruswami, 2003]. We construct an explicit binary ((1-τ)n̅,L) erasure list-decodable code having rate τ^(1+γ) (for any constant γ > 0 and small τ) and list-size poly(log 1/τ), answering simultaneously both questions, and exhibiting an explicit non-linear code that provably beats the best possible linear code. The binary erasure list-decoding problem is equivalent to the construction of explicit, low-error, strong dispersers outputting one bit with minimal entropy-loss and seed-length. For error ε, no prior explicit construction achieved seed-length better than 2log(1/ε) or entropy-loss smaller than 2log(1/ε), which are the best possible parameters for extractors. We explicitly construct an ε-error one-bit strong disperser with near-optimal seed-length (1+γ)log(1/ε) and entropy-loss O(log log1/ε). The main ingredient in our construction is a new (and almost-optimal) unbalanced two-source extractor. The extractor extracts one bit with constant error from two independent sources, where one source has length n and tiny min-entropy O(log log n) and the other source has length O(log n) and arbitrarily small constant min-entropy rate. When instantiated as a balanced two-source extractor, it improves upon Raz’s extractor [Raz, 2005] in the constant error regime. The construction incorporates recent components and ideas from extractor theory with a delicate and novel analysis needed in order to solve dependency and error issues that prevented previous papers (such as [Li, 2015; Chattopadhyay and Zuckerman, 2019; Cohen, 2016]) from achieving the above results.

Subject Classification

ACM Subject Classification
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Error-correcting codes
  • Theory of computation → Pseudorandomness and derandomization
  • Dispersers
  • Erasure codes
  • List decoding
  • Ramsey graphs
  • Two-source extractors


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  1. HL Abbott. Lower bounds for some Ramsey numbers. Discrete Mathematics, 2(4):289-293, 1972. Google Scholar
  2. Noga Alon. The Shannon capacity of a union. Combinatorica, 18(3):301-310, 1998. Google Scholar
  3. Noga Alon, Oded Goldreich, and Yishay Mansour. Almost k-wise independence versus k-wise independence. Information Processing Letters, 88(3):107-110, 2003. Google Scholar
  4. Boaz Barak. A simple explicit construction of an n^õ(log n)-Ramsey graph. arXiv preprint, 2006. URL:
  5. Boaz Barak, Guy Kindler, Ronen Shaltiel, Benny Sudakov, and Avi Wigderson. Simulating independence: New constructions of condensers, Ramsey graphs, dispersers, and extractors. Journal of the ACM (JACM), 57(4):20, 2010. Google Scholar
  6. Boaz Barak, Anup Rao, Ronen Shaltiel, and Avi Wigderson. 2-source dispersers for n^o(1) entropy, and Ramsey graphs beating the frankl-wilson construction. Annals of Mathematics, 176(3):1483-1544, 2012. Google Scholar
  7. Avraham Ben-Aroya, Dean Doron, and Amnon Ta-Shma. An efficient reduction from two-source to nonmalleable extractors: achieving near-logarithmic min-entropy. SIAM Journal on Computing, pages STOC17-31-STOC17-49, 2019. Google Scholar
  8. Jean Bourgain. More on the sum-product phenomenon in prime fields and its applications. International Journal of Number Theory, 1(01):1-32, 2005. Google Scholar
  9. Eshan Chattopadhyay and David Zuckerman. Explicit two-source extractors and resilient functions. Annals of Mathematics, 189(3):653-705, 2019. Google Scholar
  10. Benny Chor and Oded Goldreich. Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM Journal on Computing, 17(2):230-261, 1988. Google Scholar
  11. Fan RK Chung. A note on constructive methods for Ramsey numbers. Journal of Graph Theory, 5(1):109-113, 1981. Google Scholar
  12. Gil Cohen. Local correlation breakers and applications to three-source extractors and mergers. SIAM Journal on Computing, 45(4):1297-1338, 2016. Google Scholar
  13. Gil Cohen. Non-malleable extractors-new tools and improved constructions. In LIPIcs-Leibniz International Proceedings in Informatics, volume 50. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  14. Gil Cohen. Two-source dispersers for polylogarithmic entropy and improved Ramsey graphs. In Proceedings of the 48th Annual ACM Symposium on Theory of Computing (STOC 2016), pages 278-284, 2016. Google Scholar
  15. Gil Cohen. Two-source extractors for quasi-logarithmic min-entropy and improved privacy amplification protocols. In Electronic Colloquium on Computational Complexity (ECCC), volume 23, page 114, 2016. Google Scholar
  16. Zeev Dvir, Swastik Kopparty, Shubhangi Saraf, and Madhu Sudan. Extensions to the method of multiplicities, with applications to kakeya sets and mergers. SIAM Journal on Computing, 42(6):2305-2328, 2013. Google Scholar
  17. Zeev Dvir and Avi Wigderson. Kakeya sets, new mergers, and old extractors. SIAM Journal on Computing, 40(3):778-792, 2011. Google Scholar
  18. Paul Erdös. Some remarks on the theory of graphs. Bulletin of the American Mathematical Society, 53(4):292-294, 1947. Google Scholar
  19. Peter Frankl. A constructive lower bound for Ramsey numbers. Ars Combinatoria, 3(297-302):28, 1977. Google Scholar
  20. Peter Frankl and Richard M. Wilson. Intersection theorems with geometric consequences. Combinatorica, 1(4):357-368, 1981. Google Scholar
  21. Parikshit Gopalan. Constructing Ramsey graphs from boolean function representations. In Proceedings of the 21st Annual IEEE Conference on Computational Complexity (CCC 2006), pages 14-pp, 2006. Google Scholar
  22. Ronen Gradwohl, Guy Kindler, Omer Reingold, and Amnon Ta-Shma. On the error parameter of dispersers. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 294-305. Springer, 2005. Google Scholar
  23. Venkatesan Guruswami. List decoding of error-correcting codes. PhD thesis, Massachusetts Institute of Technology, 2001. Google Scholar
  24. Venkatesan Guruswami. List decoding from erasures: Bounds and code constructions. IEEE Transactions on Information Theory, 49(11):2826-2833, 2003. Google Scholar
  25. Venkatesan Guruswami. Better extractors for better codes? In Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC 2004), pages 436-444, 2004. Google Scholar
  26. Venkatesan Guruswami and Piotr Indyk. Near-optimal linear-time codes for unique decoding and new list-decodable codes over smaller alphabets. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC 2002), pages 812-821, 2002. Google Scholar
  27. Xin Li. Three-source extractors for polylogarithmic min-entropy. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2015), pages 863-882. IEEE, 2015. Google Scholar
  28. Xin Li. Improved non-malleable extractors, non-malleable codes and independent source extractors. In Proceedings of the 49th Annual ACM Symposium on Theory of Computing (STOC 2017), pages 1144-1156, 2017. Google Scholar
  29. Xin Li. Non-malleable extractors and non-malleable codes: Partially optimal constructions. In Proceedings of the 34th Computational Complexity Conference (CCC 2019). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019. Google Scholar
  30. Shachar Lovett. Recent advances on the log-rank conjecture in communication complexity. Bulletin of EATCS, 1(112), 2014. Google Scholar
  31. Shachar Lovett. Communication is bounded by root of rank. Journal of the ACM (JACM), 63(1):1, 2016. Google Scholar
  32. Raghu Meka. Explicit resilient functions matching Ajtai-Linial. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pages 1132-1148, 2017. Google Scholar
  33. Raghu Meka, Omer Reingold, and Yuan Zhou. Deterministic coupon collection and better strong dispersers. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2014. Google Scholar
  34. Zs Nagy. A constructive estimation of the Ramsey numbers. Mat. Lapok, 23:301-302, 1975. Google Scholar
  35. Moni Naor. Constructing Ramsey graphs from small probability spaces. IBM Research Report RJ, 8810, 1992. Google Scholar
  36. Noam Nisan and Avi Wigderson. On rank vs. communication complexity. Combinatorica, 15(4):557-565, 1995. Google Scholar
  37. Jaikumar Radhakrishnan and Amnon Ta-Shma. Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM Journal on Discrete Mathematics, 13(1):2-24, 2000. Google Scholar
  38. FP Ramsey. On a problem of formal logic. Proceedings of the London Mathematical Society, 2(1):264-286, 1930. Google Scholar
  39. Ran Raz. Extractors with weak random seeds. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC 2005), pages 11-20, 2005. Google Scholar
  40. Ronen Shaltiel. Recent developments in explicit constructions of extractors. Bulletin of the EATCS, 77(67-95):10, 2002. Google Scholar
  41. Ronen Shaltiel. An introduction to randomness extractors. In International Colloquium on Automata, Languages, and Programming, pages 21-41. Springer, 2011. Google Scholar
  42. Luca Trevisan. Extractors and pseudorandom generators. Journal of the ACM (JACM), 48(4):860-879, 2001. Google Scholar
  43. Avi Wigderson. Randomness extractors - applications and constructions. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2009. Google Scholar
  44. David Zuckerman. Simulating BPP using a general weak random source. Algorithmica, 16(4):367-391, 1996. Google Scholar
  45. David Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing, 3:103-128, 2007. Google Scholar
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