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# Bounds for List-Decoding and List-Recovery of Random Linear Codes

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## Cite As

Venkatesan Guruswami, Ray Li, Jonathan Mosheiff, Nicolas Resch, Shashwat Silas, and Mary Wootters. Bounds for List-Decoding and List-Recovery of Random Linear Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.9

## Abstract

A family of error-correcting codes is list-decodable from error fraction p if, for every code in the family, the number of codewords in any Hamming ball of fractional radius p is less than some integer L that is independent of the code length. It is said to be list-recoverable for input list size 𝓁 if for every sufficiently large subset of codewords (of size L or more), there is a coordinate where the codewords take more than 𝓁 values. The parameter L is said to be the "list size" in either case. The capacity, i.e., the largest possible rate for these notions as the list size L → ∞, is known to be 1-h_q(p) for list-decoding, and 1-log_q 𝓁 for list-recovery, where q is the alphabet size of the code family. In this work, we study the list size of random linear codes for both list-decoding and list-recovery as the rate approaches capacity. We show the following claims hold with high probability over the choice of the code (below q is the alphabet size, and ε > 0 is the gap to capacity). - A random linear code of rate 1 - log_q(𝓁) - ε requires list size L ≥ 𝓁^{Ω(1/ε)} for list-recovery from input list size 𝓁. This is surprisingly in contrast to completely random codes, where L = O(𝓁/ε) suffices w.h.p. - A random linear code of rate 1 - h_q(p) - ε requires list size L ≥ ⌊ {h_q(p)/ε+0.99}⌋ for list-decoding from error fraction p, when ε is sufficiently small. - A random binary linear code of rate 1 - h₂(p) - ε is list-decodable from average error fraction p with list size with L ≤ ⌊ {h₂(p)/ε}⌋ + 2. (The average error version measures the average Hamming distance of the codewords from the center of the Hamming ball, instead of the maximum distance as in list-decoding.) The second and third results together precisely pin down the list sizes for binary random linear codes for both list-decoding and average-radius list-decoding to three possible values. Our lower bounds follow by exhibiting an explicit subset of codewords so that this subset - or some symbol-wise permutation of it - lies in a random linear code with high probability. This uses a recent characterization of (Mosheiff, Resch, Ron-Zewi, Silas, Wootters, 2019) of configurations of codewords that are contained in random linear codes. Our upper bound follows from a refinement of the techniques of (Guruswami, Håstad, Sudan, Zuckerman, 2002) and strengthens a previous result of (Li, Wootters, 2018), which applied to list-decoding rather than average-radius list-decoding.

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Coding theory
##### Keywords
• list-decoding
• list-recovery
• random linear codes
• coding theory

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## References

1. Dimitris Achlioptas and Assaf Naor. The two possible values of the chromatic number of a random graph. Annals of Mathematics, 162(3):1335-1351, 2005.
2. Noga Alon and Michael Krivelevich. The concentration of the chromatic number of random graphs. Combinatorica, 17(3):303-313, 1997.
3. Eli Ben-Sasson, Swastik Kopparty, and Jaikumar Radhakrishnan. Subspace polynomials and limits to list decoding of reed-solomon codes. IEEE Transactions on Information Theory, 56(1):113-120, 2009.
4. Vladimir M Blinovsky. Code bounds for multiple packings over a nonbinary finite alphabet. Problems of Information Transmission, 41(1):23-32, 2005.
5. Volodia M. Blinovsky. Bounds for codes in the case of list decoding of finite volume. Problems of Information Transmission, 22(1):7–19, 1986.
6. Béla Bollobás and Paul Erdös. Cliques in random graphs. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 80, pages 419-427. Cambridge University Press, 1976.
7. Mahdi Cheraghchi, Venkatesan Guruswami, and Ameya Velingker. Restricted isometry of fourier matrices and list decodability of random linear codes. In Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, New Orleans, Louisiana, USA, January 6-8, 2013, pages 432-442, 2013. URL: https://doi.org/10.1137/1.9781611973105.31.
8. Dean Doron, Dana Moshkovitz, Justin Oh, and David Zuckerman. Nearly optimal pseudorandomness from hardness. Technical report, ECCC preprint TR19-099, 2019.
9. Zeev Dvir and Shachar Lovett. Subspace evasive sets. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 351-358. ACM, 2012.
10. Peter Elias. List decoding for noisy channels. Wescon Convention Record, Part 2, pages 94-104, 1957.
11. Peter Elias. Error-correcting codes for list decoding. IEEE Transactions on Information Theory, 37(1):5-12, 1991.
12. Robert G. Gallager. Low-density parity-check codes. IRE Trans. Information Theory, 8(1):21-28, 1962. URL: https://doi.org/10.1109/TIT.1962.1057683.
13. Anna C Gilbert, Hung Q Ngo, Ely Porat, Atri Rudra, and Martin J Strauss. l2/l2-foreach sparse recovery with low risk. In International Colloquium on Automata, Languages, and Programming, pages 461-472. Springer, 2013.
14. Venkatesan Guruswami. List decoding from erasures: Bounds and code constructions. IEEE Transactions on Information Theory, 49(11):2826-2833, 2003.
15. Venkatesan Guruswami, Johan Håstad, and Swastik Kopparty. On the list-decodability of random linear codes. IEEE Trans. Information Theory, 57(2):718-725, 2011. URL: https://doi.org/10.1109/TIT.2010.2095170.
16. Venkatesan Guruswami, Johan Håstad, Madhu Sudan, and David Zuckerman. Combinatorial bounds for list decoding. IEEE Trans. Information Theory, 48(5):1021-1034, 2002. URL: https://doi.org/10.1109/18.995539.
17. Venkatesan Guruswami and Piotr Indyk. Expander-based constructions of efficiently decodable codes. In 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, 14-17 October 2001, Las Vegas, Nevada, USA, pages 658-667, 2001. URL: https://doi.org/10.1109/SFCS.2001.959942.
18. 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 thiry-fourth annual ACM symposium on Theory of computing, pages 812-821, 2002.
19. Venkatesan Guruswami and Piotr Indyk. Linear time encodable and list decodable codes. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 126-135, 2003.
20. Venkatesan Guruswami and Piotr Indyk. Efficiently decodable codes meeting gilbert-varshamov bound for low rates. In SODA, volume 4, pages 756-757. Citeseer, 2004.
21. Venkatesan Guruswami and Swastik Kopparty. Explicit subspace designs. Combinatorica, 36(2):161-185, 2016.
22. Venkatesan Guruswami and Srivatsan Narayanan. Combinatorial limitations of average-radius list-decoding. IEEE Trans. Information Theory, 60(10):5827-5842, 2014. URL: https://doi.org/10.1109/TIT.2014.2343224.
23. Venkatesan Guruswami and Atri Rudra. Limits to list decoding reed-solomon codes. In Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pages 602-609, 2005.
24. Venkatesan Guruswami and Atri Rudra. Concatenated codes can achieve list-decoding capacity. Electronic Colloquium on Computational Complexity (ECCC), 15(054), 2008. URL: http://eccc.hpi-web.de/eccc-reports/2008/TR08-054/index.html.
25. Venkatesan Guruswami and Salil P. Vadhan. A lower bound on list size for list decoding. In Approximation, Randomization and Combinatorial Optimization, Algorithms and Techniques, 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005 and 9th InternationalWorkshop on Randomization and Computation, RANDOM 2005, Berkeley, CA, USA, August 22-24, 2005, Proceedings, pages 318-329, 2005. URL: https://doi.org/10.1007/11538462_27.
26. Venkatesan Guruswami and Chaoping Xing. Folded codes from function field towers and improved optimal rate list decoding. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 339-350. ACM, 2012.
27. Venkatesan Guruswami and Chaoping Xing. List decoding Reed-Solomon, Algebraic-Geometric, and Gabidulin subcodes up to the Singleton bound. In Proceedings of the forty-fifth annual ACM symposium on Theory of computing, pages 843-852. ACM, 2013.
28. Iftach Haitner, Yuval Ishai, Eran Omri, and Ronen Shaltiel. Parallel hashing via list recoverability. In Annual Cryptology Conference, pages 173-190. Springer, 2015.
29. Brett Hemenway, Noga Ron-Zewi, and Mary Wootters. Local list recovery of high-rate tensor codes & applications. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 204-215. IEEE, 2017.
30. Brett Hemenway and Mary Wootters. Linear-time list recovery of high-rate expander codes. Information and Computation, 261:202-218, 2018.
31. Piotr Indyk, Hung Q Ngo, and Atri Rudra. Efficiently decodable non-adaptive group testing. In Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms, pages 1126-1142. SIAM, 2010.
32. Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, and Shashwat Silas. On list recovery of high-rate tensor codes. Electronic Colloquium on Computational Complexity (ECCC), 2019. URL: https://eccc.weizmann.ac.il/report/2019/080/.
33. Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, and Mary Wootters. Improved decoding of folded reed-solomon and multiplicity codes. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 212-223. IEEE, 2018.
34. Ray Li and Mary Wootters. Improved list-decodability of random linear binary codes. arXiv preprint, 2018. URL: http://arxiv.org/abs/1801.07839.
35. Tomasz Luczak. A note on the sharp concentration of the chromatic number of random graphs. Combinatorica, 11(3):295-297, 1991.
36. David W Matula. Employee party problem. In Notices of the American Mathematical Society, volume 19, pages A382-A382. Amer Mathematical Soc 201 Charles St, Providence, RI 02940-2213, 1972.
37. Jonathan Mosheiff, Nicolas Resch, Noga Ron-Zewi, Shashwat Silas, and Mary Wootters. Ldpc codes achieve list decoding capacity. arXiv preprint, 2019. URL: http://arxiv.org/abs/1909.06430.
38. Hung Q Ngo, Ely Porat, and Atri Rudra. Efficiently decodable error-correcting list disjunct matrices and applications. In International Colloquium on Automata, Languages, and Programming, pages 557-568. Springer, 2011.
39. Oliver Riordan and Nicholas Wormald. The diameter of sparse random graphs. Combinatorics, Probability and Computing, 19(5-6):835-926, 2010.
40. Atri Rudra and Mary Wootters. Every list-decodable code for high noise has abundant near-optimal rate puncturings. In Proceedings of the forty-sixth annual ACM symposium on Theory of computing, pages 764-773. ACM, 2014.
41. Atri Rudra and Mary Wootters. It'll probably work out: improved list-decoding through random operations. Electronic Colloquium on Computational Complexity (ECCC), 21:104, 2014. URL: http://eccc.hpi-web.de/report/2014/104.
42. Atri Rudra and Mary Wootters. Average-radius list-recovery of random linear codes. In Proceedings of the 2018 ACM-SIAM Symposium on Discrete Algorithms, SODA, 2018.
43. Mary Wootters. On the list decodability of random linear codes with large error rates. In Symposium on Theory of Computing Conference, STOC'13, Palo Alto, CA, USA, June 1-4, 2013, pages 853-860, 2013. URL: https://doi.org/10.1145/2488608.2488716.
44. Jack Wozencraft. List decoding. Quarter Progress Report, 48:90-95, 1958.
45. Victor Vasilievich Zyablov and Mark Semenovich Pinsker. List concatenated decoding. Problemy Peredachi Informatsii, 17(4):29-33, 1981.
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