Abstract
We present a simple deterministic gap-preserving reduction from SAT to the Minimum Distance of Code Problem over \(\mathbb{F}_2\). We also show how to extend the reduction to work over any finite field (of constant size). Previously a randomized reduction was known due to Dumer, Micciancio, and Sudan [9], which was recently derandomized by Cheng and Wan [7, 8]. These reductions rely on highly non-trivial coding theoretic constructions whereas our reduction is elementary.
As an additional feature, our reduction gives a constant factor hardness even for asymptotically good codes, i.e., having constant rate and relative distance. Previously it was not known how to achieve deterministic reductions for such codes.
Research done while first author at New York University supported by NSF Expeditions grant CCF-0832795. Second author supported by NSF CAREER grant CCF-0833228, NSF Expeditions grant CCF-0832795, and BSF grant 2008059.
This is a preview of subscription content, log in via an institution to check access.
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
Ajtai, M.: The shortest vector problem in L2 is NP-hard for randomized reductions. In: Proc. 30th ACM Symposium on the Theory of Computing, pp. 10–19 (1998)
Arora, S., Lund, C., Motawani, 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)
Austrin, P., Khot, S.: A Simple Deterministic Reduction for the Gap Minimum Distance of Code Problem. arXiv report 1010.1481
Bogdanov, A., Viola, E.: Pseudorandom bits for polynomials. SIAM J. Comput. 39(6), 2464–2486 (2010)
Cai, J., Nerurkar, A.: Approximating the SVP to within a factor ( 1 + 1/dimε) is NP-hard under randomized reductions. Journal of Computer and Systems Sciences 59(2), 221–239 (1999)
Cheng, Q., Wan, D.: Complexity of decoding positive-rate reed-solomon codes. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 283–293. Springer, Heidelberg (2008)
Cheng, Q., Wan, D.: A deterministic reduction for the gap minimum distance problem. In: Proceedings of the ACM Symposium on the Theory of Computing, pp. 33–38 (2009)
Dumer, I., Micciancio, D., Sudan, M.: Hardness of approximating the minimum distance of a linear code. In: Proc. 40th IEEE Symposium on Foundations of Computer Science (1999)
Feige, U., Goldwasser, S., Lovász, L., Safra, S., Szegedy, M.: Interactive proofs and the hardness of approximating cliques. Journal of the ACM 43(2), 268–292 (1996)
Haviv, I., Regev, O.: Tensor-based hardness of the shortest vector problem to within almost polynomial factors. In: Proceedings of the ACM Symposium on the Theory of Computing, pp. 469–477 (2007)
Khot, S.: Hardness of approximating the shortest vector problem in high L p norms. In: Proc. 44th IEEE Symposium on Foundations of Computer Science (2003)
Khot, S.: Hardness of approximating the shortest vector problem in lattices. In: Proc. 45th IEEE Symposium on Foundations of Computer Science, pp. 126–135 (2004)
Lovett, S.: Unconditional pseudorandom generators for low degree polynomials. Theory of Computing 5(1), 69–82 (2009)
Micciancio, D.: The shortest vector problem is NP-hard to approximate to within some constant. SIAM Journal on Computing 30(6), 2008–2035 (2000)
Vardy, A.: The intractability of computing the minimum distance of a code. IEEE Transactions on Information Theory 43(6), 1757–1766 (1997)
Viola, E.: The sum of d small-bias generators fools polynomials of degree d. Computational Complexity 18(2), 209–217 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Austrin, P., Khot, S. (2011). A Simple Deterministic Reduction for the Gap Minimum Distance of Code Problem. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_40
Download citation
DOI: https://doi.org/10.1007/978-3-642-22006-7_40
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22005-0
Online ISBN: 978-3-642-22006-7
eBook Packages: Computer ScienceComputer Science (R0)