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Sparser Abelian High Dimensional Expanders

Authors Yotam Dikstein , Siqi Liu , Avi Wigderson

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  • Theory of computation
  • Mathematics of computing → Spectra of graphs
Keywords
  • Local spectral expander
  • coboundary expander
  • Grassmannian expander

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Abstract

The focus of this paper is the development of new elementary techniques for the construction and analysis of high dimensional expanders. Specifically, we present two new explicit constructions of Cayley high dimensional expanders (HDXs) over the abelian group 𝔽₂ⁿ. Our expansion proofs use only linear algebra and combinatorial arguments.
The first construction gives local spectral HDXs of any constant dimension and subpolynomial degree exp(n^ε) for every ε > 0, improving on a construction by Golowich [Golowich, 2023] which achieves ε = 1/2. [Golowich, 2023] derives these HDXs by sparsifying the complete Grassmann poset of subspaces. The novelty in our construction is the ability to sparsify any expanding Grassmann posets, leading to iterated sparsification and much smaller degrees. The sparse Grassmannian (which is of independent interest in the theory of HDXs) serves as the generating set of the Cayley graph.
Our second construction gives a 2-dimensional HDX of any polynomial degree exp(ε n) for any constant ε > 0, which is simultaneously a spectral expander and a coboundary expander. To the best of our knowledge, this is the first such non-trivial construction. We name it the Johnson complex, as it is derived from the classical Johnson scheme, whose vertices serve as the generating set of this Cayley graph. This construction may be viewed as a derandomization of the recent random geometric complexes of [Liu et al., 2023]. Establishing coboundary expansion through Gromov’s "cone method" and the associated isoperimetric inequalities is the most intricate aspect of this construction.
While these two constructions are quite different, we show that they both share a common structure, resembling the intersection patterns of vectors in the Hadamard code. We propose a general framework of such "Hadamard-like" constructions in the hope that it will yield new HDXs.

Cite As Get BibTex

Yotam Dikstein, Siqi Liu, and Avi Wigderson. Sparser Abelian High Dimensional Expanders. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 7:1-7:98, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.7

Author Details

Yotam Dikstein
  • Institute for Advanced Study, Princeton, NJ, USA
Siqi Liu
  • Institute for Advanced Study, Princeton, NJ, USA
Avi Wigderson
  • Institute for Advanced Study, Princeton, NJ, USA

Funding

  • Dikstein, Yotam: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1926686.
  • Liu, Siqi: Supported by Minerva Research Foundation Member Fund. This work was partially conducted at the Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) with support from DIMACS and Rutgers University.
  • Wigderson, Avi: Supported by NSF grant CCF-1900460.

Acknowledgements

We thank Louis Golowich for helpful discussions and Gil Melnik for assistance with the figures.

References

  1. Vedat Alev, Fernando Jeronimo, and Madhur Tulsiani. Approximating constraint satisfaction problems on high-dimensional expanders. In Proc. 59th IEEE Symp. on Foundations of Computer Science, pages 180-201, November 2019. URL: https://doi.org/10.1109/FOCS.2019.00021.
  2. Noga Alon, Oded Goldreich, Johan Håstad, and René Peralta. Simple constructions of almost k-wise independent random variables. Random Structures & Algorithms, 3(3):289-304, 1992. URL: https://doi.org/10.1002/RSA.3240030308.
  3. Noga Alon and Yuval Roichman. Random cayley graphs and expanders. Random Structures & Algorithms, 5(2):271-284, 1994. URL: https://doi.org/10.1002/RSA.3240050203.
  4. Nima Anari, Vishesh Jain, Frederic Koehler, Huy Tuan Pham, and Thuy-Duong Vuong. Entropic independence i: Modified log-sobolev inequalities for fractionally log-concave distributions and high-temperature ising models. arXiv preprint, 2021. URL: https://arxiv.org/abs/2106.04105.
  5. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials ii: high-dimensional walks and an fpras for counting bases of a matroid. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 1-12, 2019. URL: https://doi.org/10.1145/3313276.3316385.
  6. Anurag Anshu, Nikolas P Breuckmann, and Chinmay Nirkhe. Nlts hamiltonians from good quantum codes. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 1090-1096, 2023. URL: https://doi.org/10.1145/3564246.3585114.
  7. Sanjeev Arora and Madhu Sudan. Improved low degree testing and its applications. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 485-495, El Paso, Texas, 1997. URL: https://doi.org/10.1145/258533.258642.
  8. Mitali Bafna, Max Hopkins, Tali Kaufman, and Shachar Lovett. Hypercontractivity on high dimensional expanders. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 185-194, 2022. URL: https://doi.org/10.1145/3519935.3520040.
  9. Mitali Bafna, Noam Lifshitz, and Dor Minzer. Constant degree direct product testers with small soundness, 2024. URL: https://arxiv.org/abs/2402.00850.
  10. Mitali Bafna and Dor Minzer. Characterizing direct product testing via coboundary expansion. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 1978-1989, 2024. URL: https://doi.org/10.1145/3618260.3649714.
  11. Mitali Bafna, Dor Minzer, and Nikhil Vyas. Quasi-linear size pcps with small soundness from hdx. arXiv preprint, 2024. URL: https://doi.org/10.48550/arXiv.2407.12762.
  12. Cristina M Ballantine. Ramanujan type buildings. Canadian Journal of Mathematics, 52(6):1121-1148, 2000. Google Scholar
  13. Imre Bárány. A generalization of carathéodory’s theorem. Discrete Mathematics, 40(2-3):141-152, 1982. URL: https://doi.org/10.1016/0012-365X(82)90115-7.
  14. Yonatan Bilu and Nathan Linial. Constructing expander graphs by 2-lifts and discrepancy vs. spectral gap. In 45th Annual IEEE Symposium on Foundations of Computer Science, pages 404-412. IEEE, 2004. URL: https://doi.org/10.1109/FOCS.2004.19.
  15. Endre Boros and Zoltán Füredi. The number of triangles covering the center of an n-set. Geometriae Dedicata, 17:69-77, 1984. Google Scholar
  16. Donald I Cartwright, Patrick Solé, and Andrzej Żuk. Ramanujan geometries of type an. Discrete mathematics, 269(1-3):35-43, 2003. Google Scholar
  17. Michael Chapman, Nathan Linial, and Peled Yuval. Expander graphs — both local and global. Combinatorica, 40:473-509, 2020. URL: https://doi.org/10.1007/s00493-019-4127-8.
  18. Michael Chapman and Alexander Lubotzky. Stability of homomorphisms, coverings and cocycles i: Equivalence, 2023. URL: https://arxiv.org/abs/2310.17474.
  19. Michael Chapman and Alexander Lubotzky. Stability of homomorphisms, coverings and cocycles ii: Examples, applications and open problems, 2023. URL: https://arxiv.org/abs/2311.06706.
  20. Zongchen Chen, Andreas Galanis, Daniel Štefankovič, and Eric Vigoda. Rapid mixing for colorings via spectral independence. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1548-1557. SIAM, 2021. Google Scholar
  21. Fan Chung and Paul Horn. The spectral gap of a random subgraph of a graph. Internet Mathematics, 4(2-3):225-244, 2007. URL: https://doi.org/10.1080/15427951.2007.10129296.
  22. David Conlon. Hypergraph expanders from cayley graphs. Israel Journal of Mathematics, 233(1):49-65, 2019. Google Scholar
  23. David Conlon, Jonathan Tidor, and Yufei Zhao. Hypergraph expanders of all uniformities from cayley graphs. Proceedings of the London Mathematical Society, 121(5):1311-1336, 2020. Google Scholar
  24. Philippe Delsarte and Vladimir I. Levenshtein. Association schemes and coding theory. IEEE Transactions on Information Theory, 44(6):2477-2504, 1998. URL: https://doi.org/10.1109/18.720545.
  25. Yotam Dikstein. New high dimensional expanders from covers. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pages 826-838, 2023. URL: https://doi.org/10.1145/3564246.3585183.
  26. Yotam Dikstein and Irit Dinur. Agreement testing theorems on layered set systems. In Proceedings of the 60th Annual IEEE Symposium on Foundations of Computer Science, 2019. Google Scholar
  27. Yotam Dikstein and Irit Dinur. Agreement theorems for high dimensional expanders in the small soundness regime: the role of covers. CoRR, abs/2308.09582, 2023. URL: https://doi.org/10.48550/arXiv.2308.09582.
  28. Yotam Dikstein and Irit Dinur. Coboundary and cosystolic expansion without dependence on dimension or degree. CoRR, abs/2304.01608, 2023. URL: https://doi.org/10.48550/arXiv.2304.01608.
  29. Yotam Dikstein and Irit Dinur. Swap cosystolic expansion. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 1956-1966, 2024. URL: https://doi.org/10.1145/3618260.3649780.
  30. Yotam Dikstein, Irit Dinur, Yuval Filmus, and Prahladh Harsha. Boolean functions on high-dimensional expanders. arXiv preprint arXiv:1804.08155, 2018. URL: https://arxiv.org/abs/1804.08155.
  31. Yotam Dikstein, Irit Dinur, and Alexander Lubotzky. Low acceptance agreement tests via bounded-degree symplectic hdxs, 2024. URL: https://eccc.weizmann.ac.il/report/2024/019/.
  32. Yotam Dikstein and Max Hopkins. Chernoff bounds and reverse hypercontractivity on hdx. arXiv preprint arXiv:2404.10961, 2024. URL: https://doi.org/10.48550/arXiv.2404.10961.
  33. Irit Dinur. The pcp theorem by gap amplification. Journal of the ACM (JACM), 54(3):12-es, 2007. Google Scholar
  34. Irit Dinur, Shai Evra, Ron Livne, Alexander Lubotzky, and Shahar Mozes. Good locally testable codes. arXiv preprint arXiv:2207.11929, 2022. URL: https://doi.org/10.48550/arXiv.2207.11929.
  35. Irit Dinur, Yuval Filmus, Prahladh Harsha, and Madhur Tulsiani. Explicit sos lower bounds from high-dimensional expanders. In 12th Innovations in Theoretical Computer Science (ITCS 2021), 2021. URL: https://arxiv.org/abs/2009.05218.
  36. Irit Dinur and Elazar Goldenberg. Locally testing direct products in the low error range. In Proc. 49th IEEE Symp. on Foundations of Computer Science, 2008. Google Scholar
  37. Irit Dinur, Prahladh Harsha, Tali Kaufman, Inbal Livni Navon, and Amnon Ta-Shma. List-decoding with double samplers. SIAM journal on computing, 50(2):301-349, 2021. URL: https://doi.org/10.1137/19M1276650.
  38. Irit Dinur and Tali Kaufman. High dimensional expanders imply agreement expanders. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 974-985. IEEE, 2017. URL: https://doi.org/10.1109/FOCS.2017.94.
  39. Irit Dinur, Siqi Liu, and Rachel Yun Zhang. New codes on high dimensional expanders. arXiv preprint arXiv:2308.15563, 2023. URL: https://doi.org/10.48550/arXiv.2308.15563.
  40. Irit Dinur and Inbal Livni Navon. Exponentially small soundness for the direct product z-test. In 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, pages 29:1-29:50, 2017. URL: https://doi.org/10.4230/LIPICS.CCC.2017.29.
  41. Irit Dinur and Roy Meshulam. Near coverings and cosystolic expansion. Archiv der Mathematik, 118(5):549-561, May 2022. URL: https://doi.org/10.1007/s00013-022-01720-6.
  42. Dominic Dotterrer and Matthew Kahle. Coboundary expanders. Journal of Topology and Analysis, 4(04):499-514, 2012. Google Scholar
  43. Dominic Dotterrer, Tali Kaufman, and Uli Wagner. On expansion and topological overlap. Geometriae Dedicata, 195:307-317, 2018. Google Scholar
  44. Shai Evra and Tali Kaufman. Bounded degree cosystolic expanders of every dimension. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 36-48, 2016. URL: https://doi.org/10.1145/2897518.2897543.
  45. Ofer Gabber and Zvi Galil. Explicit constructions of linear-sized superconcentrators. Journal of Computer and System Sciences, 22(3):407-420, 1981. URL: https://doi.org/10.1016/0022-0000(81)90040-4.
  46. Jason Gaitonde, Max Hopkins, Tali Kaufman, Shachar Lovett, and Ruizhe Zhang. Eigenstripping, spectral decay, and edge-expansion on posets. arXiv preprint arXiv:2205.00644, 2022. URL: https://doi.org/10.48550/arXiv.2205.00644.
  47. Howard Garland. p-adic curvature and the cohomology of discrete subgroups of p-adic groups. Ann. of Math., 97(3):375-423, 1973. URL: https://doi.org/10.2307/1970829.
  48. Oded Goldreich and Shmuel Safra. A combinatorial consistency lemma with application to proving the PCP theorem. In RANDOM: International Workshop on Randomization and Approximation Techniques in Computer Science. LNCS, 1997. Google Scholar
  49. Louis Golowich. Improved Product-Based High-Dimensional Expanders. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021), volume 207, 2021. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.38.
  50. Louis Golowich. From grassmannian to simplicial high-dimensional expanders. arXiv preprint arXiv:2305.02512, 2023. URL: https://doi.org/10.48550/arXiv.2305.02512.
  51. Roy Gotlib and Tali Kaufman. List agreement expansion from coboundary expansion, 2022. URL: https://arxiv.org/abs/2210.15714.
  52. M. Gromov. Singularities, expanders and topology of maps. part 2: from combinatorics to topology via algebraic isoperimetry. Geom. Funct. Anal., 20:416-526, 2010. URL: https://doi.org/10.1007/s00039-010-0073-8.
  53. Anna Gundert and Uli Wagner. On eigenvalues of random complexes. Israel Journal of Mathematics, 216:545-582, 2016. Google Scholar
  54. Tom Gur, Noam Lifshitz, and Siqi Liu. Hypercontractivity on high dimensional expanders. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 176-184, 2022. URL: https://doi.org/10.1145/3519935.3520004.
  55. Christopher Hoffman, Matthew Kahle, and Elliot Paquette. The threshold for integer homology in random d-complexes. Discrete & Computational Geometry, 57:810-823, 2017. URL: https://doi.org/10.1007/S00454-017-9863-1.
  56. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bulletin of the American Mathematical Society, 43(4):439-561, 2006. Google Scholar
  57. Max Hopkins and Ting-Chun Lin. Explicit lower bounds against ω(n)-rounds of sum-of-squares. arXiv preprint arXiv:2204.11469, 2022. URL: https://doi.org/10.48550/arXiv.2204.11469.
  58. Russell Impagliazzo, Valentine Kabanets, and Avi Wigderson. New direct-product testers and 2-query PCPs. SIAM J. Comput., 41(6):1722-1768, 2012. URL: https://doi.org/10.1137/09077299X.
  59. Fernando Granha Jeronimo, Shashank Srivastava, and Madhur Tulsiani. Near-linear time decoding of ta-shma’s codes via splittable regularity. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1527-1536, 2021. URL: https://doi.org/10.1145/3406325.3451126.
  60. Tali Kaufman, David Kazhdan, and Alexander Lubotzky. Isoperimetric inequalities for ramanujan complexes and topological expanders. Geometric and Functional Analysis, 26(1):250-287, 2016. Google Scholar
  61. Tali Kaufman and Alexander Lubotzky. High dimensional expanders and property testing. In Innovations in Theoretical Computer Science, ITCS'14, Princeton, NJ, USA, January 12-14, 2014, pages 501-506, 2014. URL: https://doi.org/10.1145/2554797.2554842.
  62. Tali Kaufman and David Mass. High dimensional random walks and colorful expansion. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  63. Tali Kaufman and David Mass. Good distance lattices from high dimensional expanders. (manuscript), 2018. URL: https://arxiv.org/abs/1803.02849.
  64. Tali Kaufman and David Mass. Unique-neighbor-like expansion and group-independent cosystolic expansion. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  65. Tali Kaufman and David Mass. Double balanced sets in high dimensional expanders. arXiv preprint arXiv:2211.09485, 2022. URL: https://doi.org/10.48550/arXiv.2211.09485.
  66. Tali Kaufman and Izhar Oppenheim. Construction of new local spectral high dimensional expanders. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 773-786, 2018. URL: https://doi.org/10.1145/3188745.3188782.
  67. Tali Kaufman and Izhar Oppenheim. High order random walks: Beyond spectral gap. Combinatorica, 40:245-281, 2020. URL: https://doi.org/10.1007/S00493-019-3847-0.
  68. Tali Kaufman and Izhar Oppenheim. Coboundary and cosystolic expansion from strong symmetry. In 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, pages 84:1-84:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.84.
  69. Tali Kaufman and Ran J Tessler. Garland’s technique for posets and high dimensional grassmannian expanders. arXiv preprint, 2021. URL: https://arxiv.org/abs/2101.12621.
  70. Mikhail Koshelev. Spectrum of johnson graphs. Discrete Mathematics, 346(3):113262, 2023. URL: https://doi.org/10.1016/J.DISC.2022.113262.
  71. Dmitry N Kozlov and Roy Meshulam. Quantitative aspects of acyclicity. Research in the Mathematical Sciences, 6(4):1-32, 2019. Google Scholar
  72. Wen-Ching Winnie Li. Ramanujan hypergraphs. Geometric & Functional Analysis GAFA, 14:380-399, 2004. Google Scholar
  73. Nathan Linial and Roy Meshulam. Homological connectivity of random 2-complexes. Combinatorica, 26(4):475-487, 2006. URL: https://doi.org/10.1007/S00493-006-0027-9.
  74. Siqi Liu, Sidhanth Mohanty, Tselil Schramm, and Elizabeth Yang. Local and global expansion in random geometric graphs. In Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, pages 817-825, New York, NY, USA, 2023. Association for Computing Machinery. URL: https://doi.org/10.1145/3564246.3585106.
  75. Siqi Liu, Sidhanth Mohanty, and Elizabeth Yang. High-dimensional expanders from expanders. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020), volume 151, page 12. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  76. Alexander Lubotzky. Expander graphs in pure and applied mathematics. Bulletin of the American Mathematical Society, 49(1):113-162, 2012. Google Scholar
  77. Alexander Lubotzky, Roy Meshulam, and Shahar Mozes. Expansion of building-like complexes. Groups, Geometry, and Dynamics, 10(1):155-175, 2016. Google Scholar
  78. Alexander Lubotzky, Ralph Phillips, and Peter Sarnak. Ramanujan graphs. Combinatorica, 8(3):261-277, 1988. URL: https://doi.org/10.1007/BF02126799.
  79. Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Explicit constructions of ramanujan complexes of type ad. European Journal of Combinatorics, 26(6):965-993, 2005. URL: https://doi.org/10.1016/J.EJC.2004.06.007.
  80. Roger C Lyndon, Paul E Schupp, RC Lyndon, and PE Schupp. Combinatorial group theory, volume 89. Springer, 1977. Google Scholar
  81. Neal Madras and Dana Randall. Markov chain decomposition for convergence rate analysis. Annals of Applied Probability, pages 581-606, 2002. Google Scholar
  82. Adam W. Marcus, Daniel A. Spielman, and Nikhil Srivastava. Interlacing families i: Bipartite ramanujan graphs of all degrees. Annals of Mathematics, 182:307-325, 2015. URL: https://doi.org/10.4007/annals.2015.182.1.7.
  83. Adam W Marcus, Daniel A Spielman, and Nikhil Srivastava. Interlacing families ii: Mixed characteristic polynomials and the Kadison-Singer problem. Annals of Mathematics, pages 327-350, 2015. Google Scholar
  84. Grigorii Aleksandrovich Margulis. Explicit constructions of concentrators. Problemy Peredachi Informatsii, 9(4):71-80, 1973. Google Scholar
  85. Roy Meshulam and Nathan Wallach. Homological connectivity of random k-dimensional complexes. Random Structures & Algorithms, 34(3):408-417, 2009. URL: https://doi.org/10.1002/RSA.20238.
  86. Ryan O'Donnell and Kevin Pratt. High-dimensional expanders from chevalley groups. In Shachar Lovett, editor, 37th Computational Complexity Conference, CCC 2022, July 20-23, 2022, Philadelphia, PA, USA, volume 234 of LIPIcs, pages 18:1-18:26. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.18.
  87. Izhar Oppenheim. Local spectral expansion approach to high dimensional expanders part i: Descent of spectral gaps. Discrete & Computational Geometry, 59(2):293-330, 2018. URL: https://doi.org/10.1007/S00454-017-9948-X.
  88. 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.
  89. Mark S Pinsker. On the complexity of a concentrator. In 7th International Telegraffic Conference, volume 4, pages 1-318, 1973. Google Scholar
  90. Ran Raz. A parallel repetition theorem. SIAM Journal on Computing, 27(3):763-803, June 1998. URL: https://doi.org/10.1137/S0097539795280895.
  91. Ran Raz and Shmuel Safra. 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, STOC '97, pages 475-484, New York, NY, USA, 1997. ACM. URL: https://doi.org/10.1145/258533.258641.
  92. Omer Reingold. Undirected connectivity in log-space. J. ACM, 55(4), 2008. URL: https://doi.org/10.1145/1391289.1391291.
  93. Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 3-13. IEEE, 2000. URL: https://doi.org/10.1109/SFCS.2000.892006.
  94. 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.
  95. David Surowski. Covers of simplicial complexes and applications to geometry. Geometriae Dedicata, 16:35-62, April 1984. URL: https://doi.org/10.1007/BF00147420.
  96. Amnon Ta-Shma. Explicit, almost optimal, epsilon-balanced codes. In Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pages 238-251, 2017. URL: https://doi.org/10.1145/3055399.3055408.
  97. Egbert R Van Kampen. On some lemmas in the theory of groups. American Journal of Mathematics, 55(1):268-273, 1933. Google Scholar
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