Eli Ben-Sasson
ABSTRACT
We describe new constructions of error correcting codes, obtained by "degree-lifting" a short algebraic geometry base-code of block-length q to a lifted-code of block-length qm, for arbitrary integer m. The construction generalizes the way degree-d, univariate polynomials evaluated over the q-element field (also known as Reed-Solomon codes) are "lifted" to degree-d, m-variate polynomials (Reed-Muller codes). A number of properties are established: The rate of the degree-lifted code is approximately a 1/m!-fraction of the rate of the base-code. The relative distance of the degree-lifted code is at least as large as that of the base-code. This is proved using a generalization of the Schwartz-Zippel Lemma to degree-lifted Algebraic-Geometry codes. [Local correction] If the base code is invariant under a group that is "close" to being doubly-transitive (in a precise manner defined later then the degree-lifted code is locally correctable with query complexity at most q2. The automorphisms of the base-code are crucially used to generate query-sets, abstracting the use of affine-lines in the local correction procedure of Reed-Muller codes. Taking a concrete illustrating example, we show that degree-lifted Hermitian codes form a family of locally correctable codes over an alphabet that is significantly smaller than that obtained by Reed-Muller codes of similar constant rate, message length, and distance.
- N. Alon, T. Kaufman, M. Krivelevich, S. Litsyn, and D. Ron. Testing Reed-Muller codes. IEEE Transactions on Information Theory, 51(11):4032--4039, 2005. Google Scholar
Digital Library
- S. Arora and S. Safra. Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM, 45(1):70--122, Jan. 1998. Google ScholarDigital Library
- S. Arora and M. Sudan. Improved low-degree testing and its applications. Combinatorica, 23(3):365--426, 2003.Google ScholarCross Ref
- L. Babai, L. Fortnow, L. A. Levin, and M. Szegedy. Checking computations in polylogarithmic time. In STOC '91: Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, pages 21--32, New York, NY, USA, 1991. ACM. Google ScholarDigital Library
- L. Babai, L. Fortnow, and C. Lund. Addendum to non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 2:374, 1992.Google ScholarDigital Library
- L. Babai, L. Fortnow, N. Nisan, and A. Wigderson. Bpp has subexponential time simulations unless exptime has publishable proofs. Computational Complexity, 3:307--318, 1993. Google ScholarDigital Library
- L. Babai, A. Shpilka, and D. Stefankovic. Locally testable cyclic codes. In IEEE, editor, Proceedings: 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003, 11--14 October 2003, Cambridge, Massachusetts, pages 116--125, pub-IEEE:adr, 2003. IEEE Computer Society Press. Google ScholarDigital Library
- D. Beaver and J. Feigenbaum. Hiding instances in multioracle queries. In Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science, STACS '90, pages 37--48. 1990. Google ScholarDigital Library
- A. Ben-Aroya, K. Efremenko, and A. Ta-Shma. Local list decoding with a constant number of queries. Electronic Colloquium on Computational Complexity (ECCC), 17:47, 2010.Google Scholar
- A. Ben-Aroya and A. Ta-Shma. Constructing small-bias sets from algebraic-geometric codes. In FOCS, pages 191--197. IEEE Computer Society, 2009. Google ScholarDigital Library
- E. Ben-Sasson, E. Grigorescu, G. Maatouk, A. Shpilka, and M. Sudan. On sums of locally testable affine invariant properties. Electronic Colloquium on Computational Complexity (ECCC), 18:79, 2011.Google Scholar
- E. Ben-Sasson, V. Guruswami, T. Kaufman, M. Sudan, and M. Viderman. Locally testable codes require redundant testers. In Proceedings of the 24th Annual IEEE Conference on Computational Complexity (CCC), pages 52--61, 2009. Google ScholarDigital Library
- E. Ben-Sasson, P. Harsha, and S. Raskhodnikova. Some 3CNF properties are hard to test. SIAM J. on Computing, 35(1):1--21, 2005. Google ScholarDigital Library
- E. Ben-Sasson, G. Maatouk, A. Shpilka, and M. Sudan. Symmetric LDPC codes are not necessarily locally testable. In preparation, 2010.Google Scholar
- E. Ben-Sasson, N. Ron-Zewi, and M. Sudan. Sparse affine-invariant linear codes are locally testable. Electronic Colloquium on Computational Complexity (ECCC), 19:49, 2012.Google Scholar
- E. Ben-Sasson and M. Sudan. Simple PCPs with poly-log rate and query complexity. In H. N. Gabow and R. Fagin, editors, Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, USA, May 22--24, 2005, pages 266--275. ACM, 2005. Google ScholarDigital Library
- E. Ben-Sasson and M. Sudan. Limits on the rate of locally testable affine-invariant codes. Electronic Colloquium on Computational Complexity (ECCC), 17:108, 2010.Google Scholar
- E. Ben-Sasson and M. Viderman. Composition of Semi-LTCs by Two-Wise Tensor Products. In Proceedings of the Approximation, Randomization, and Combinatorial Optimization, (APPROX-RANDOM 2009), volume 5687 of Lecture Notes in Computer Science, pages 378--391. Springer, 2009. Google ScholarDigital Library
- E. Ben-Sasson and M. Viderman. Tensor Products of Weakly Smooth Codes are Robust. Theory of Computing, 5(1):239--255, 2009.Google ScholarCross Ref
- Y. M. Chee, T. Feng, S. Ling, H. Wang, and L. F. Zhang. Query-efficient locally decodable codes of subexponential length. CoRR, abs/1008.1617, 2010.Google Scholar
- B. Chor, O. Goldreich, E. Kushilevitz, and M. Sudan. Private information retrieval. Journal of the ACM, 45:965--981, 1998. Google ScholarDigital Library
- D. Coppersmith and A. Rudra. On the Robust Testability of Product of Codes. Electronic Colloquium on Computational Complexity (ECCC), (104), 2005.Google Scholar
- A. Deshpande, R. Jain, T. Kavitha, J. Radhakrishnan, and S. V. Lokam. Better lower bounds for locally decodable codes. In IEEE Conference on Computational Complexity, pages 184--193, 2002. Google ScholarDigital Library
- I. Dinur, M. Sudan, and A. Wigderson. Robust local testability of tensor products of LDPC codes. In J. Díaz, K. Jansen, J. D. P. Rolim, and U. Zwick, editors, APPROX-RANDOM, volume 4110 of Lecture Notes in Computer Science, pages 304--315. Springer, 2006. Google ScholarDigital Library
- Z. Dvir, P. Gopalan, and S. Yekhanin. Matching vector codes. Electronic Colloquium on Computational Complexity (ECCC), 17:12, 2010.Google Scholar
- K. Efremenko. 3-query locally decodable codes of subexponential length. In M. Mitzenmacher, editor, STOC, pages 39--44. ACM, 2009. Google ScholarDigital Library
- K. Efremenko. From irreducible representations to locally decodable codes. In STOC, pages 327--338, 2012. Google ScholarDigital Library
- L. Fortnow and S. P. Vadhan, editors. Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6--8 June 2011. ACM, 2011. Google Scholar
- K. Friedl and M. Sudan. Some improvements to total degree tests. In ISTCS, pages 190--198, 1995. Google ScholarDigital Library
- A. Garcia and H. Stichtenoth. On the Asymptotic Behaviour of Some Towers of Function Fields over Finite Fields. Journal of Number Theory, 61:248--273, 1996.Google ScholarCross Ref
- O. Goldreich, H. J. Karloff, L. J. Schulman, and L. Trevisan. Lower bounds for linear locally decodable codes and private information retrieval. In IEEE Conference on Computational Complexity, pages 175--183, 2002. Google ScholarDigital Library
- V. Goppa. Algebraic-geometric codes. Izu. Akad. Nauk SSSR Ser. Mat, 46(4):762--781, 1982.Google Scholar
- V. D. Goppa. Geometry and Codes. Springer, 1988.Google ScholarCross Ref
- E. Grigorescu, T. Kaufman, and M. Sudan. Succinct representation of codes with applications to testing. In I. Dinur, K. Jansen, J. Naor, and J. D. P. Rolim, editors, APPROX-RANDOM, volume 5687 of Lecture Notes in Computer Science, pages 534--547. Springer, 2009. Google ScholarDigital Library
- A. Guo, S. Kopparty, and M. Sudan. New affine-invariant codes from lifting. In ITCS, pages 529--540, 2013. Google ScholarDigital Library
- R. W. Hamming. Error detecting and error correcing codes. Bell System Technical Journal, 29:147---160, 1950.Google ScholarCross Ref
- J. P. Hansen and H. Stichtenoth. Group codes on certain algebraic curves with many rational points. Appl. Algebra Eng. Commun. Comput., 1:67--77, 1990.Google ScholarDigital Library
- S. Hansen. Error-correcting codes from higher-dimensional varieties. Finite Fields Appl., 7:530--552, 2001. Google ScholarDigital Library
- R. Impagliazzo and A. Wigderson. P = BPP if e requires exponential circuits: Derandomizing the xor lemma. In F. T. Leighton and P. W. Shor, editors, STOC, pages 220--229. ACM, 1997. Google ScholarDigital Library
- T. Itoh and Y. Suzuki. Improved constructions for query-efficient locally decodable codes of subexponential length. IEICE Transactions, 93-D(2):263--270, 2010.Google Scholar
- J. Katz and L. Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In Proceedings of the thirty-second annual ACM symposium on Theory of computing, STOC '00, pages 80--86, New York, NY, USA, 2000. ACM. Google ScholarDigital Library
- T. Kaufman and A. Lubotzky. Edge transitive ramanujan graphs and symmetric ldpc good codes. In H. J. Karloff and T. Pitassi, editors, STOC, pages 359--366. ACM, 2012. Google ScholarDigital Library
- T. Kaufman and M. Sudan. Algebraic property testing: the role of invariance. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC), pages 403--412, 2008. Google ScholarDigital Library
- T. Kaufman and M. Viderman. Locally testable vs. locally decodable codes. In M. J. Serna, R. Shaltiel, K. Jansen, and J. D. P. Rolim, editors, APPROX-RANDOM, volume 6302 of Lecture Notes in Computer Science, pages 670--682. Springer, 2010. Google ScholarDigital Library
- T. Kaufman and A. Wigderson. Symmetric ldpc codes and local testing. In A. C.-C. Yao, editor, ICS, pages 406--421. Tsinghua University Press, 2010.Google Scholar
- I. Kerenidis and R. de Wolf. Exponential lower bound for 2-query locally decodable codes via a quantum argument. J. Comput. Syst. Sci., 69(3):395--420, 2004. Google ScholarDigital Library
- S. Kopparty, S. Saraf, and S. Yekhanin. High-rate codes with sublinear-time decoding. In Fortnow and Vadhancite DBLP: conf/stoc/2011, pages 167--176. Google ScholarDigital Library
- J. L. L. Gold and H. Schenck. Cayley-bacharach and evaluation codes on complete intersections. J. Pure Appl. Algebra, 196:91--99, 2005.Google ScholarCross Ref
- G. Lachaud. Number of points of plane sections and linear codes defined on algebraic varieties. Arithmetic, Geometry and Coding Theory, Proceedings Luminy, pages 77--104, 1993.Google Scholar
- R. J. Lipton. Efficient checking of computations. In Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science, STACS '90, pages 207--215, 1990. Google ScholarDigital Library
- J. B. Little. Algebraic geometry codes from higher dimensional varieties. CoRR, abs/0802.2349, 2008.Google Scholar
- C. Lund, L. Fortnow, H. Karloff, and N. Noam. Algebraic methods for interactive proof systems. Journal of the ACM, 39(4):859--868, 1992. Google ScholarDigital Library
- S. V. M.A. Tsfasman and T. Zink. Modular curves, shimura curves, and goppa codes, better than varshamov-gilbert bound. math. Nachr., 109:21--28, 1982.Google ScholarCross Ref
- F. J. MacWilliams and N. J. A. Sloane. The theory of error-correcting codes. North-Holland Amsterdam, 1978.Google Scholar
- K. Obata. Optimal lower bounds for 2-query locally decodable linear codes. In J. D. P. Rolim and S. P. Vadhan, editors, RANDOM, volume 2483 of Lecture Notes in Computer Science, pages 39--50. Springer, 2002. Google ScholarDigital Library
- J. Pedersen. A function field related to the ree group. Lect. Notes Math., 1518:122--132, 1992.Google ScholarCross Ref
- P. Raghavendra. A note on yekhanin's locally decodable codes. Electronic Colloquium on Computational Complexity (ECCC), 14(016), 2007.Google Scholar
- A. A. Razborov and S. Yekhanin. An omega(n1/3) lower bound for bilinear group based private information retrieval. Theory of Computing, 3(1):221--238, 2007.Google ScholarCross Ref
- I. S. Reed. A class of multiple-error-correcting codes and the decoding scheme. IEEE Transactions on Information Theory, (4):38--49, 1954.Google Scholar
- F. rodier. Codes from flag varieties over a finite field. J. Pure Appl. Algebra, 178:203--214, 2003.Google ScholarCross Ref
- R. Shaltiel and C. Umans. Simple extractors for all min-entropies and a new pseudorandom generator. J. ACM, 52(2):172--216, 2005. Google ScholarDigital Library
- A. Shamir. IP = PSPACE. Journal of the ACM, 39(4):869--877, 1992. Google ScholarDigital Library
- C. E. Shannon. A mathematical theory of communication. Bell System Technical Journal, 27:379---423, 623--656, 1948.Google ScholarCross Ref
- C. E. Shannon. Communication theory - exposition of fundamentals. IEEE Transactions on Information Theory, 1:44--47, 1953.Google Scholar
- H. Stichtenoth. On automorphisms of geometric goppa codes. Journal of Algebra, page 113.Google Scholar
- H. Stichtenoth. Algebraic function fields and codes. Universitext. Springer, 1993.Google Scholar
- M. Sudan. Invariance in property testing. Electronic Colloquium on Computational Complexity (ECCC), (051), 2010.Google Scholar
- M. Sudan, L. Trevisan, and S. P. Vadhan. Pseudorandom generators without the xor lemma (extended abstract). In J. S. Vitter, L. L. Larmore, and F. T. Leighton, editors, STOC, pages 537--546. ACM, 1999. Google ScholarDigital Library
- M. Tsfasman and S.G.Vladut. Algebraic-geometric codes. (Kluwer, Dordrecht, 1991. Google ScholarDigital Library
- P. Valiant. The tensor product of two codes is not necessarily robustly testable. In C. Chekuri, K. Jansen, J. D. P. Rolim, and L. Trevisan, editors, APPROX-RANDOM, volume 3624 of Lecture Notes in Computer Science, pages 472--481. Springer, 2005. Google ScholarDigital Library
- S. Wehner and R. de Wolf. Improved lower bounds for locally decodable codes and private information retrieval. In L. Caires, G. F. Italiano, L. Monteiro, C. Palamidessi, and M. Yung, editors, ICALP, volume 3580 of Lecture Notes in Computer Science, pages 1424--1436. Springer, 2005. Google ScholarDigital Library
- D. P. Woodruff. New lower bounds for general locally decodable codes. Electronic Colloquium on Computational Complexity (ECCC), 14(006), 2007.Google Scholar
- C. Xing. On automorphism groups of the hermitian codes. IEEE Transactions on Information Theory, 41(6):1629--1635, 1995. Google ScholarCross Ref
- S. Yekhanin. Towards 3-query locally decodable codes of subexponential length. J. ACM, 55(1), 2008. Google ScholarDigital Library
- S. Yekhanin. Locally decodable codes: A brief survey. In Y. M. Chee, Z. Guo, S. Ling, F. Shao, Y. Tang, H. Wang, and C. Xing, editors, IWCC, volume 6639 of Lecture Notes in Computer Science, pages 273--282. Springer, 2011. Google ScholarDigital Library
Index Terms
- A new family of locally correctable codes based on degree-lifted algebraic geometry codes
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