Abstract
There has been significant interest lately in the task of constructing codes that are testable with a small number of random probes. Ben-Sasson and Sudan show that the repeated tensor product of codes leads to a general class of locally testable codes. One question that is not settled by their work is the local testability of a code generated by a single application of the tensor product. Special cases of this question have been studied in the literature in the form of “tests for bivariate polynomials”, where the tensor product has been shown to be locally testable for certain families of codes. However the question remained open for the tensor product of generic families of codes. Here we resolve the question negatively, giving families of codes whose tensor product does not have good local testability properties.
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References
Arora, S.: Probabilistic checking of proofs and the hardness of approximation problems. PhD thesis, University of California at Berkeley (1994)
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)
Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM 45(1), 70–122 (1998)
Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.: Robust PCPs of proximity, shorter PCPs and applications to coding. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 1–10 (2004)
Ben-Sasson, E., Sudan, M.: Robust Locally Testable Codes and Products of Codes. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 286–297. Springer, Heidelberg (2004)
Ben-Sasson, E., Sudan, M.: Simple PCPs with Poly-log Rate and Query Complexity. In: 37th STOC (2005) (to appear)
Friedl, K., Sudan, M.: Some improvements to total degree tests. In: Proceedings of the 3rd Annual Israel Symposium on Theory of Computing and Systems, Tel Aviv, Israel, January 4-6, pp. 190–198 (1995), available online at http://theory.csail.mit.edu/~madhu/papers/friedl.ps
Goldreich, O., Sudan, M.: Locally testable codes and PCPs of almostlinear length. In: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, Vancouver, Canada, November 16-19 (2002)
MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. Elsevier/North-Holland (1981)
Polishchuk, A., Spielman, D.A.: Nearly linear-size holographic proofs. In: Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, Montreal, Quebec, Canada, May 23-25, pp. 194–203 (1994)
Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing 25(2), 252–271 (1996)
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 29th STOC, pp. 475–484 (1997)
Sudan, M.: Algorithmic introduction to coding theory. Lecture notes (2001), available from http://theory.csail.mit.edu/~madhu/FT01/
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Valiant, P. (2005). The Tensor Product of Two Codes Is Not Necessarily Robustly Testable. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds) Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2005 2005. Lecture Notes in Computer Science, vol 3624. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11538462_40
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DOI: https://doi.org/10.1007/11538462_40
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