Abstract
We show that expander codes, when properly instantiated, are high-rate list recoverable codes with linear-time list recovery algorithms. List recoverable codes have been useful recently in constructing efficiently list-decodable codes, as well as explicit constructions of matrices for compressive sensing and group testing. Previous list recoverable codes with linear-time decoding algorithms have all had rate at most \(1/2\); in contrast, our codes can have rate \(1 - \varepsilon \) for any \(\varepsilon > 0\). We can plug our high-rate codes into a framework of Alon and Luby (1996) and Meir (2014) to obtain linear-time list recoverable codes of arbitrary rates \(R\), which approach the optimal trade-off between the number of non-trivial lists provided and the rate of the code.
While list-recovery is interesting on its own, our primary motivation is applications to list-decoding. A slight strengthening of our result would imply linear-time and optimally list-decodable codes for all rates. Thus, our result is a step in the direction of solving this important problem.
Mary Wootterso—Research funded by NSF MSPRF grant DMS-1400558
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References
Alon, N., Luby, M.: A linear time erasure-resilient code with nearly optimal recovery. IEEE Transactions on Information Theory 42(6), 1732–1736 (1996)
Barg, A., Zemor, G.: Error exponents of expander codes. IEEE Transactions on Information Theory 48(6), 1725–1729 (2002)
Barg, A., Zemor, G.: Concatenated codes: serial and parallel. IEEE Transactions on Information Theory 51(5), 1625–1634 (2005)
Barg, A., Zemor, G.: Distance properties of expander codes. IEEE Transactions on Information Theory 52(1), 78–90 (2006)
Dvir, Z., Lovett, S.: Subspace evasive sets. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC), pp. 351–358. ACM (2012)
Gallager, R.G.: Low Density Parity-Check Codes. Technical report. MIT (1963)
Gilbert, A.C., Ngo, H.Q., Porat, E., Rudra, A., Strauss, M.J.: \(\ell \) \(_\text{2 }\)/\(\ell \) \(_\text{2 }\)-foreach sparse recovery with low risk. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 461–472. Springer, Heidelberg (2013)
Guruswami, V.: List decoding from erasures: Bounds and code constructions. IEEE Transactions on Information Theory 49(11), 2826–2833 (2003)
Guruswami, V.: List decoding of error-correcting codes. LNCS, vol. 3282. Springer, Heidelberg (2004)
Guruswami, V.: Linear-algebraic list decoding of folded reed-solomon codes. In: Proceedings of the 26th Annual Conference on Computational Complexity (CCC), pp. 77–85. IEEE (2011)
Guruswami, V., Indyk, P:. Expander-based constructions of efficiently decodable codes. In: Proceedings of the 42nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 658–667. IEEE (October 2001)
Guruswami, V., Indyk, P.: Near-optimal linear-time codes for unique decoding and new list-decodable codes over smaller alphabets. In: Proceedings of the 34th Annual ACM Aymposium on Theory of computing (STOC), pp. 812–821. ACM (2002)
Guruswami, V., Indyk, P.: Linear time encodable and list decodable codes. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC), pp. 126–135. ACM, New York (2003)
Guruswami, V., Indyk, P.: Linear-time list decoding in error-free settings. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 695–707. Springer, Heidelberg (2004)
Guruswami, V., Kopparty, S.: Explicit subspace designs. In: Proceedings of the 54th Annual IEEE Symposium on Foundations of Computing (FOCS), pp. 608–617. IEEE (2013)
Guruswami, V., Rudra, A.: Explicit codes achieving list decoding capacity: Error-correction with optimal redundancy. IEEE Transactions on Information Theory 54(1), 135–150 (2008)
Guruswami, V., Sudan, M.: Improved decoding of Reed-Solomon and algebraic-geometry codes. IEEE Transactions on Information Theory 45(6) (1999)
Guruswami, V., Xing, C.: Folded codes from function field towers and improved optimal rate list decoding. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC), pp. 339–350. ACM (2012)
Guruswami, V., Xing, C.: List decoding reed-solomon, algebraic-geometric, and gabidulin subcodes up to the singleton bound. In: Proceedings of the 45th Annual ACM Symposium on Theory of Computing (STOC), pp. 843–852. ACM (2013)
Hemenway, B., Ostrovsky, R., Wootters, M.: Local correctability of expander codes. Information and Computation (2014)
Hemenway, B., Wootters, M.: Linear-time list recovery of high-rate expander codes. ArXiv preprint 1503.01955 (2015)
Hoory, S., Linial, N., Wigderson, A.: Expander graphs and their applications. Bulletin of the American Mathematical Society 43(4), 439–561 (2006)
Indyk, P., Ngo, H.Q., Rudra, A.: Efficiently decodable non-adaptive group testing. In: Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1126–1142. Society for Industrial and Applied Mathematics (2010)
Lubotzky, A., Phillips, R., Sarnak, P.: Ramanujan graphs. Combinatorica 8(3), 261–277 (1988)
Margulis, G.A.: Explicit Group-Theoretical Constructions of Combinatorial Schemes and Their Application to the Design of Expanders and Concentrators. Probl. Peredachi Inf. 24(1), 51–60 (1988)
Meir, O.: Locally correctable and testable codes approaching the singleton bound, ECCC Report TR14-107 (2014)
Morgenstern, M.: 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)
Ngo, H.Q., Porat, E., Rudra, A.: Efficiently decodable compressed sensing by list-recoverable codes and recursion. In: Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS), vol. 14, pp. 230–241 (2012)
Sipser, M., Spielman, D.A.: Expander codes. IEEE Transactions in Information Theory 42(6) (1996)
Tanner, R.: A recursive approach to low complexity codes. IEEE Transactions on Information Theory 27(5), 533–547 (1981)
Zemor, G.: On expander codes. IEEE Transactions on Information Theory 47(2), 835–837 (2001)
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Hemenway, B., Wootters, M. (2015). Linear-Time List Recovery of High-Rate Expander Codes. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_57
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