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We expose a strong connection between good $2$-query locally testable codes (LTCs) and high dimensional expanders.
Towards local testability for quantum coding
Anthony Leverrier, Vivien Londe, and Gilles Zémor · 1911
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Endliche Erzeugbarkeit arithmetischer Gruppen über Funktionenkörpern
Helmut Behr · 1969
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Minkowskische Reduktionstheorie über Funktionenkörpern
G. Harder · 1969
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Groupes réductifs sur un corps local
F. Bruhat and J. Tits · 1972
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On normal subgroups of Chevalley groups over commutative rings
Eiichi Abe and Kazuo Suzuki · 1976
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On the congruence subgroup problem
Madabusi S. Raghunathan · 1976
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Cobounded subgroups in algebraic groups over local fields
G. A. Margulis · 1977
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Strong approximation for semi-simple groups over function fields
Gopal Prasad · 1977
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Reductive groups over local fields
Jacques Tits · 1979
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Introduction to affine group schemes
William C. Waterhouse · 1979
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Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits
Gopal Prasad · 1982
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Finite presentability of arithmetic groups over global function fields
Helmut Behr · 1985
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Cohomology of sheaves
Birger Iversen · 1986
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On the congruence subgroup problem. II
M. S. Raghunathan · 1986
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Algebraic groups and number theory
Vladimir Platonov and Andrei Rapinchuk · 1991
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Sheaves in geometry and logic
Saunders Mac Lane and Ieke Moerdijk · 1992
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Some improvements to total degree tests
Katalin Friedl and Madhu Sudan · 1995
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Computation of the metaplectic kernel
Gopal Prasad and Andrei S. Rapinchuk · 1996
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Robust characterizations of polynomials with applications to program testing
Ronitt Rubinfeld and Madhu Sudan · 1996
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Expander codes
Michael Sipser and Daniel A. Spielman · 1996
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Ramanujan local systems on graphs
Bruce W. Jordan and Ron Livné · 1997
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Valuation-like maps and the congruence subgroup property
Andrei S. Rapinchuk and Yoav Segev · 2001
Cited alongside, same era.
Lattice chain models for affine buildings of classical type
Peter Abramenko and Gabriele Nebe · 2002
Cited alongside, same era.
Bounds on 2-query codeword testing
Eli Ben-Sasson, Oded Goldreich, and Madhu Sudan · 2003
Cited alongside, same era.
Ramanujan geometries of type A ~ n \tilde{A}_{n}
Donald I. Cartwright, Patrick Solé, and Andrzej Żuk · 2003
Cited alongside, same era.
Ramanujan hypergraphs
W.-C. W. Li · 2004
Cited alongside, same era.
Explicit constructions of Ramanujan complexes of type A ~ d \tilde{A}_{d}
Alexander Lubotzky, Beth Samuels, and Uzi Vishne · 2005
Cited alongside, same era.
Relative commutator calculus in Chevalley groups
Roozbeh Hazrat, Nikolai Vavilov, and Zuhong Zhang · 2013
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High dimensional expanders and property testing
Tali Kaufman and Alexander Lubotzky · 2014
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Sheaves on graphs, their homological invariants, and a proof of the Hanna Neumann conjecture: with an appendix by Warren Dicks
Joel Friedman · 2015
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Vanishing of cohomology and property (T) for groups acting on weighted simplicial complexes
Izhar Oppenheim · 2015
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The ramanujan property for simplicial complexes, 2016
Uriya A. First · 2016
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Isoperimetric inequalities for Ramanujan complexes and topological expanders
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Ramanujan complexes of type A ~ d \tilde{A}_{d}
Alexander Lubotzky, Beth Samuels, and Uzi Vishne · 2005
Cited alongside, same era.
Homological connectivity of random 2-complexes
Nathan Linial and Roy Meshulam · 2006
Cited alongside, same era.
The Margulis-Platonov conjecture for SL 1 , D {\rm SL}_{1,D} and 2-generation of finite simple groups
Andrei S. Rapinchuk · 2006
Cited alongside, same era.
The PCP theorem by gap amplification
Irit Dinur · 2007
Cited alongside, same era.
Explicit construction of a Ramanujan ( n 1 , n 2 , … , n d − 1 ) (n_{1},n_{2},\dots,n_{d-1}) -regular hypergraph
Alireza Sarveniazi · 2007
Cited alongside, same era.
Buildings
Peter Abramenko and Kenneth S. Brown · 2008
Cited alongside, same era.
Tali Kaufman, David Kazhdan, and Alexander Lubotzky · 2016
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Bounds on 2-query locally testable codes with affine tests
Gillat Kol and Ran Raz · 2016
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Expansion of building-like complexes
Alexander Lubotzky, Roy Meshulam, and Shahar Mozes · 2016
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Bounded degree cosystolic expanders of every dimension
Shai Evra and Tali Kaufman · 2017
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On expansion and topological overlap
Dominic Dotterrer, Tali Kaufman, and Uli Wagner · 2018
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Cosystolic expanders over any abelian group
Tali Kaufman and David Mass · 2018
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Construction of new local spectral high dimensional expanders
Tali Kaufman and Izhar Oppenheim · 2018
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Graph codes and local systems, 2018
Roy Meshulam · 2018
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Locally testable codes via high-dimensional expanders
Yotam Dikstein, Irit Dinur, Prahladh Harsha, and Noga Ron-Zewi · 2020
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Stacks Project
The Stacks Project Authors · 2020
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Locally testable codes with constant rate, distance, and locality, 2021
Irit Dinur, Shai Evra, Ron Livne, Alexander Lubotzky, and Shahar Mozes · 2021
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Weighted mixing lemmas and coboundary expansion of sheaves on graphs
Uriya A. First and Tali Kaufman · 2021
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High dimensional expansion implies amplified local testability
Tali Kaufman and Izhar Oppenheim · 2021
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Asymptotically good quantum and locally testable classical LDPC codes, 2021
Pavel Panteleev and Gleb Kalachev · 2021
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Congruence subgroups: group schemes versus embeddings in GL n \mathrm{GL}_{n}
Uriya A. First · 2022
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