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Testing Tensor Products of Algebraic Codes

Authors Sumegha Garg , Madhu Sudan , Gabriel Wu

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ACM Subject Classification
  • Mathematics of computing → Coding theory
Keywords
  • Algebraic geometry codes
  • Robust testability
  • Tensor products of codes

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Abstract

Motivated by recent advances in locally testable codes and quantum LDPCs based on robust testability of tensor product codes, we explore the local testability of tensor products of (an abstraction of) algebraic geometry codes. Such codes are parameterized by, in addition to standard parameters such as block length n and dimension k, their genus g. We show that the tensor product of two algebraic geometry codes is robustly locally testable provided n = Ω((k+g)²). Apart from Reed-Solomon codes, this seems to be the first explicit family of two-wise tensor codes of high dual distance that is robustly locally testable by the natural test that measures the expected distance of a random row/column from the underlying code.

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Sumegha Garg, Madhu Sudan, and Gabriel Wu. Testing Tensor Products of Algebraic Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 59:1-59:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2025.59

Author Details

Sumegha Garg
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Madhu Sudan
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
Gabriel Wu
  • Harvard University, Cambridge, MA, USA

Funding

  • Garg, Sumegha: Part of this work was done when the author was at Harvard University supported by Michael O. Rabin Postdoctoral Fellowship, and at Stanford University supported by the Simons Collaboration on the theory of algorithmic fairness and Simons Foundation Investigators Award 689988.
  • Sudan, Madhu: Supported in part by a Simons Investigator Award and NSF Award CCF 2152413.

Acknowledgements

We would like to thank Venkatesan Guruswami and Swastik Kopparty for valuable conversations on AG codes, and the anonymous referees of RANDOM 2025 for their careful reading and corrections.

References

  1. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70-122, 1998. URL: https://doi.org/10.1145/273865.273901.
  2. László Babai, Lance Fortnow, Leonid A. Levin, and Mario Szegedy. Checking computations in polylogarithmic time. In Cris Koutsougeras and Jeffrey Scott Vitter, editors, Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, May 5-8, 1991, New Orleans, Louisiana, USA, pages 21-31. ACM, 1991. URL: https://doi.org/10.1145/103418.103428.
  3. Eli Ben-Sasson, Venkatesan Guruswami, Tali Kaufman, Madhu Sudan, and Michael Viderman. Locally testable codes require redundant testers. SIAM J. Comput., 39(7):3230-3247, 2010. URL: https://doi.org/10.1137/090779875.
  4. Eli Ben-Sasson and Madhu Sudan. Robust locally testable codes and products of codes. Random Struct. Algorithms, 28(4):387-402, 2006. URL: https://doi.org/10.1002/rsa.20120.
  5. Don Coppersmith and Atri Rudra. On the robust testability of product of codes. Electron. Colloquium Comput. Complex., TR05-104, 2005. URL: https://eccc.weizmann.ac.il/eccc-reports/2005/TR05-104/index.html.
  6. David A Cox. Primes of the Form x2+ ny2: Fermat, Class Field Theory, and Complex Multiplication. with Solutions, volume 387. American Mathematical Soc., 2022. Google Scholar
  7. Irit Dinur, Shai Evra, Ron Livne, Alexander Lubotzky, and Shahar Mozes. Locally testable codes with constant rate, distance, and locality. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 357-374. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520024.
  8. Irit Dinur, Min-Hsiu Hsieh, Ting-Chun Lin, and Thomas Vidick. Good quantum LDPC codes with linear time decoders. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 905-918. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585101.
  9. Irit Dinur, Madhu Sudan, and Avi Wigderson. Robust local testability of tensor products of LDPC codes. In Josep Díaz, Klaus Jansen, José D. P. Rolim, and Uri Zwick, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, 9th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2006 and 10th International Workshop on Randomization and Computation, RANDOM 2006, Barcelona, Spain, August 28-30 2006, Proceedings, volume 4110 of Lecture Notes in Computer Science, pages 304-315. Springer, 2006. URL: https://doi.org/10.1007/11830924_29.
  10. Iwan M. Duursma and Ralf Kötter. Error-locating pairs for cyclic codes. IEEE Trans. Inf. Theory, 40(4):1108-1121, 1994. URL: https://doi.org/10.1109/18.335964.
  11. Oded Goldreich and Or Meir. The tensor product of two good codes is not necessarily robustly testable. Inf. Process. Lett., 112(8-9):351-355, 2012. URL: https://doi.org/10.1016/j.ipl.2012.01.007.
  12. Venkatesan Guruswami and Madhu Sudan. On representations of algebraic-geometry codes. IEEE Trans. Inf. Theory, 47(4):1610-1613, 2001. URL: https://doi.org/10.1109/18.923745.
  13. Tom Høholdt, Jacobus H Van Lint, and Ruud Pellikaan. Algebraic geometry codes. Handbook of coding theory, 1(Part 1):871-961, 1998. Google Scholar
  14. Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. Mip*=re. CoRR, abs/2001.04383, 2020. URL: https://doi.org/10.48550/arXiv.2001.04383.
  15. Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen. Quantum soundness of testing tensor codes. CoRR, abs/2111.08131, 2021. URL: https://doi.org/10.48550/arXiv.2111.08131.
  16. Gleb Kalachev and Pavel Panteleev. Two-sided robustly testable codes. CoRR, abs/2206.09973, 2022. URL: https://doi.org/10.48550/arXiv.2206.09973.
  17. Gleb Kalachev and Pavel Panteleev. Maximally extendable product codes are good coboundary expanders. CoRR, abs/2501.01411, 2025. URL: https://doi.org/10.48550/arXiv.2501.01411.
  18. Tali Kaufman and Madhu Sudan. Algebraic property testing: the role of invariance. In Cynthia Dwork, editor, Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 403-412. ACM, 2008. URL: https://doi.org/10.1145/1374376.1374434.
  19. Anthony Leverrier and Gilles Zémor. Quantum tanner codes. In 63rd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2022, Denver, CO, USA, October 31 - November 3, 2022, pages 872-883. IEEE, 2022. URL: https://doi.org/10.1109/FOCS54457.2022.00117.
  20. Or Meir. Combinatorial construction of locally testable codes. SIAM J. Comput., 39(2):491-544, 2009. URL: https://doi.org/10.1137/080729967.
  21. Pavel Panteleev and Gleb Kalachev. Asymptotically good quantum and locally testable classical LDPC codes. In Stefano Leonardi and Anupam Gupta, editors, STOC '22: 54th Annual ACM SIGACT Symposium on Theory of Computing, Rome, Italy, June 20 - 24, 2022, pages 375-388. ACM, 2022. URL: https://doi.org/10.1145/3519935.3520017.
  22. Ruud Pellikaan. On decoding by error location and dependent sets of error positions. Discret. Math., 106-107:369-381, 1992. URL: https://doi.org/10.1016/0012-365X(92)90567-Y.
  23. Alexander Polishchuk and Daniel A. Spielman. Nearly-linear size holographic proofs. In Frank Thomson Leighton and Michael T. Goodrich, editors, Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, 23-25 May 1994, Montréal, Québec, Canada, pages 194-203. ACM, 1994. URL: https://doi.org/10.1145/195058.195132.
  24. Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM J. Comput., 25(2):252-271, 1996. URL: https://doi.org/10.1137/S0097539793255151.
  25. Henning Stichtenoth. Algebraic function fields and codes, volume 254 of Universitext. Springer, 1993. Google Scholar
  26. Paul Valiant. The tensor product of two codes is not necessarily robustly testable. In Chandra Chekuri, Klaus Jansen, José D. P. Rolim, and Luca Trevisan, editors, Approximation, Randomization and Combinatorial Optimization, Algorithms and Techniques, 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005 and 9th InternationalWorkshop on Randomization and Computation, RANDOM 2005, Berkeley, CA, USA, August 22-24, 2005, Proceedings, volume 3624 of Lecture Notes in Computer Science, pages 472-481. Springer, 2005. URL: https://doi.org/10.1007/11538462_40.
  27. Michael Viderman. Strong ltcs with inverse poly-log rate and constant soundness. In 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, 26-29 October, 2013, Berkeley, CA, USA, pages 330-339. IEEE Computer Society, 2013. URL: https://doi.org/10.1109/FOCS.2013.43.
  28. Michael Viderman. A combination of testability and decodability by tensor products. Random Struct. Algorithms, 46(3):572-598, 2015. URL: https://doi.org/10.1002/rsa.20498.
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