Abstract
This paper describes recent results which revolve around the question of the rate attainable by families of error correcting codes that are locally testable. Emphasis is placed on motivating the problem of proving upper bounds on the rate of these codes and a number of interesting open questions for future research are suggested.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM 45(1), 70–122 (1998)
Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the ACM 45(3), 501–555 (1998)
Dinur, I.: The PCP theorem by gap amplification. Journal of the ACM 54(3), 12:1–12:44 (2007)
Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45(4), 653–750 (1998)
Goldreich, O.: Short locally testable codes and proofs (survey). Electronic Colloquium on Computational Complexity (ECCC) (014) (2005)
Trevisan, L.: Some applications of coding theory in computational complexity. Quaderni di Matematica 13, 347–424 (2004)
MacWilliams, F., Sloane, N.: The theory of error-correcting codes. North-Holland, Amsterdam (1978)
Ben-Sasson, E., Goldreich, O., Sudan, M.: Bounds on 2-query codeword testing. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 216–227. Springer, Heidelberg (2003)
Babai, L., Shpilka, A., Stefankovic, D.: Locally testable cyclic codes. IEEE Transactions on Information Theory 51(8), 2849–2858 (2005)
Ben-Sasson, E., Guruswami, V., Kaufman, T., Sudan, M., Viderman, M.: Locally testable codes require redundant testers. In: CCC 2009: Proceedings of the 2009 24th Annual IEEE Conference on Computational Complexity, Washington, DC, USA, pp. 52–61. IEEE Computer Society, Los Alamitos (2009)
Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs, and nonapproximability—towards tight results. SIAM Journal on Computing 27(3), 804–915 (1998)
Meir, O.: On the efficiency of non-uniform pcpp verifiers. Electronic Colloquium on Computational Complexity (ECCC) 15(064) (2008)
Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. In: STOC, pp. 73–83. ACM, New York (1990)
Bellare, M., Coppersmith, D., Hastad, J., Kiwi, M., Sudan, M.: Linearity testing in characteristic two. IEEE Transactions on Information Theory 42(6), 1781–1795 (1996)
Babai, L., Fortnow, L., Levin, L., Szegedy, M.: Checking computations in polylogarithmic time. In: Proceedings of the Twenty-third Annual ACM Symposium on Theory of Computing, pp. 21–32. ACM, New York (1991)
Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of the Twenty-ninth Annual ACM Symposium on Theory of Computing, pp. 475–484. ACM, New York (1997)
Arora, S., Sudan, M.: Improved low-degree testing and its applications. Combinatorica 23(3), 365–426 (2003)
Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D.: Testing reed-muller codes. IEEE Transactions on Information Theory 51(11), 4032–4039 (2005)
Kaufman, T., Ron, D.: Testing polynomials over general fields. SIAM J. Comput. 36(3), 779–802 (2006)
Samorodnitsky, A.: Low-degree tests at large distances. In: Johnson, D.S., Feige, U. (eds.) STOC, pp. 506–515. ACM, New York (2007)
Kaufman, T., Sudan, M.: Algebraic property testing: the role of invariance. In: Ladner, R.E., Dwork, C. (eds.) STOC, pp. 403–412. ACM, New York (2008)
Grigorescu, E., Kaufman, T., Sudan, M.: 2-transitivity is insufficient for local testability. In: IEEE Conference on Computational Complexity, pp. 259–267. IEEE Computer Society, Los Alamitos (2008)
Grigorescu, E., Kaufman, T., Sudan, M.: Succinct representation of codes with applications to testing. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX–RANDOM 2009. LNCS, vol. 5687, pp. 534–547. Springer, Heidelberg (2009)
Ben-Sasson, E., Sudan, M.: Limits on the rate of locally testable affine-invariant codes. Electronic Colloquium on Computational Complexity (ECCC) (108) (2010)
Sudan, M.: Invariance in Property Testing. ECCC, TR10-051 (2010)
Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.P.: Robust PCPs of proximity, shorter PCPs, and applications to coding. SIAM J. Comput. 36(4), 889–974 (2006)
Dinur, I., Reingold, O.: Assignment testers: Towards a combinatorial proof of the PCP theorem. SIAM J. Comput. 36(4), 975–1024 (2006)
Meir, O.: Combinatorial construction of locally testable codes. SIAM J. Comput. 39(2), 491–544 (2009)
Ben-Sasson, E., Sudan, M.: Robust locally testable codes and products of codes. Random Struct. Algorithms 28(4), 387–402 (2006)
Moshkovitz, D., Raz, R.: Two-query pcp with subconstant error. J. ACM 57(5) (2010)
Dinur, I., Harsha, P.: Composition of low-error 2-query pcps using decodable pcps. In: FOCS, pp. 472–481. IEEE Computer Society, Los Alamitos (2009)
Kaufman, T., Sudan, M.: Sparse random linear codes are locally decodable and testable. In: FOCS, pp. 590–600. IEEE Computer Society, Los Alamitos (2007)
Kopparty, S., Saraf, S.: Local list-decoding and testing of random linear codes from high error. In: Schulman, L.J. (ed.) STOC, pp. 417–426. ACM, New York (2010)
Ben-Sasson, E., Viderman, M.: Low rate is insufficient for local testability. In: Shaltiel, R. (ed.) Proc. 14th Intl. Workshop on Randomization and Computation - RANDOM 2010 (September 2010)
Kilian, J.: A note on efficient zero-knowledge proofs and arguments (extended abstract). In: STOC, pp. 723–732. ACM, New York (1992)
Micali, S.: Computationally sound proofs. SIAM J. Comput. 30(4), 1253–1298 (2000)
Barak, B., Goldreich, O.: Universal arguments and their applications. SIAM J. Comput. 38(5), 1661–1694 (2008)
Ben-Sasson, E., Harsha, P., Lachish, O., Matsliah, A.: Sound 3-query PCPPs are long. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 686–697. Springer, Heidelberg (2008)
Guruswami, V.: On 2-query codeword testing with near-perfect completeness. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 267–276. Springer, Heidelberg (2006)
Kol, G., Raz, R.: Bounds on 2-Query Locally Testable Codes with Affine Tests. ECCC Report TR09-138 (2009)
Kol, G., Raz, R.: Locally testable codes analogues to the unique games conjecture do not exist. ECCC Report TR09-128 (2009)
Ben-Sasson, E., Harsha, P., Raskhodnikova, S.: Some 3CNF properties are hard to test. SIAM J. Comput. 35(1), 1–21 (2005)
Gallager, R.: Low-density parity-check codes. IRE Transactions on Information Theory 8(1), 21–28 (1962)
Sipser, M., Spielman, D.: Expander codes. IEEE Transactions on Information Theory 42(6), 1710–1722 (1996)
Spielman, D.: Computationally efficient error-correcting codes and holographic proofs. PhD thesis, MIT (1995)
Ben-Sasson, E., Viderman, M.: Dense locally testable codes have bad rate (2010) (unpublished manuscript)
Balog, A., Szemerédi, E.: A statistical theorem of set addition. Combinatorica 14(3), 263–268 (1994)
Gowers, W.T.: A new proof of szemerèdi’s theorem for arithmetic progressions of length four. Geom. Funct. Anal. 8(3), 529–551 (1998)
Freiman, G.A.: Foundations of a structural theory of set addition, vol. 37. American Mathematical Society, Providence (1973)
Ruzsa, I.Z.: An analog of freiman’s theorem in groups. Astèrique 258, 323–326 (1999)
Grigorescu, E., Kaufman, T., Sudan, M.: Succinct representation of codes with applications to testing. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pp. 534–547 (2009)
Kaufman, T., Viderman, M.: Locally testable vs. locally decodable codes. In: Shaltiel, R. (ed.) Proc. 14th Intl. Workshop on Randomization and Computation - RANDOM 2010 (2010)
Woodruff, D.: New lower bounds for general locally decodable codes. Electronic Colloquium on Computational Complexity (ECCC) 14(006) (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Ben-Sasson, E. (2010). Limitation on the Rate of Families of Locally Testable Codes. In: Goldreich, O. (eds) Property Testing. Lecture Notes in Computer Science, vol 6390. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16367-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-642-16367-8_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-16366-1
Online ISBN: 978-3-642-16367-8
eBook Packages: Computer ScienceComputer Science (R0)