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Lower Bounds for Maximally Recoverable Tensor Code and Higher Order MDS Codes [article]

Joshua Brakensiek, Sivakanth Gopi, Visu Makam
2021 arXiv   pre-print
We introduce the notion of higher order MDS codes (MDS(ℓ) codes) which is an interesting generalization of the well-known MDS codes, where ℓ captures the order of genericity of points in a low-dimensional  ...  Using this characterization and duality, we prove that (m,n,a=1,b)-MR tensor codes require fields of size q=Ω_m,b(n^min{b,m}-1). Our lower bound also extends to the setting of a>1.  ...  Acknowledgements We would like to thank Sergey Yekhanin and Venkatesan Guruswami for helpful discussions and encouraging us to work on this problem.  ... 
arXiv:2107.10822v1 fatcat:hvidhs5btjgajfsfovm56hhuo4

Maximally Recoverable Codes for Grid-like Topologies [article]

Parikshit Gopalan, Guangda Hu, Swastik Kopparty, Shubhangi Saraf, Carol Wang, Sergey Yekhanin
2016 arXiv   pre-print
We consider the field-size needed for achieving maximal recoverability: the strongest reliability possible with a given topology.  ...  in such topologies including the first super-polynomial lower bound on the field size.  ...  We do not know how to construct Maximally Recoverable tensor product codes, even for the special case of T m×n (1, b, 0) where we now know a classification of correctable failure patterns. 4.  ... 
arXiv:1605.05412v4 fatcat:tscjoanuqrexjme5pkg52vb2mu

Coding Theory for Inference, Learning and Optimization (Dagstuhl Seminar 18112)

Po-Ling Loh, Arya Mazumdar, Dimitris Papailiopoulos, Rüdiger Urbanke, Michael Wagner
2018 Dagstuhl Reports  
This report documents the program and the outcomes of Dagstuhl Seminar 18112, "Coding Theory for Inference, Learning and Optimization."  ...  The discussions and collaborations facilitated by this seminar were intended to spark new ideas about how coding theory may be used to improve and inform modern techniques for data analytics.  ...  A lower bound for maximally recoverable codes with locality Venkatesan Guruswami (Carnegie Mellon University -Pittsburgh, US) License Creative Commons BY 3.0 Unported license © Venkatesan Guruswami  ... 
doi:10.4230/dagrep.8.3.60 dblp:journals/dagstuhl-reports/LohMPU18 fatcat:j6hzkkbykjaqvl47k24orl5rlu

Holographic duality between local Hamiltonians from random tensor networks [article]

Harriet Apel, Tamara Kohler, Toby Cubitt
2021 arXiv   pre-print
for all boundary regions.  ...  We also obtain complementary recovery of local bulk operators for any boundary bipartition.  ...  The authors are grateful to Michael Walter for helpful initial discussions and pointing out a gap in the proof concerning concentration of random stabilizer tensors. T. S.  ... 
arXiv:2105.12067v3 fatcat:zcpucxsqwvendolljfcjhdsslq

Algebraic geometry codes and some applications [article]

Alain Couvreur, Hugues Randriambololona
2020 arXiv   pre-print
We summarize the major results on various problems such as: asymptotic parameters, improved estimates on the minimum distance, and decoding algorithms.  ...  This article surveys the development of the theory of algebraic geometry codes since their discovery in the late 70's.  ...  Improvement: locally recoverable codes with higher local distance.  ... 
arXiv:2009.01281v1 fatcat:2o7m2zhylnanjpljkikw4sx2ya

Optimal Quaternary (r,delta)-Locally Repairable Codes Achieving the Singleton-type Bound [article]

Kenneth W. Shum, Jie Hao
2021 arXiv   pre-print
We classify all quaternary locally repairable codes that attain the Singleton-type upper bound for minimum distance.  ...  For each combination of achievable code parameters, an explicit code construction is given.  ...  For distance larger than 4, we split the presentation into Sections VII and VIII; Section VII is for low rate and Section VIII is for higher rate.  ... 
arXiv:2112.05377v1 fatcat:zhj4abruvvazdgdgzprtpnfkli

Toy Models of Holographic Duality between local Hamiltonians [article]

Tamara Kohler, Toby Cubitt
2019 arXiv   pre-print
Holographic quantum error correcting codes (HQECC) have been proposed as toy models for the AdS/CFT correspondence, and exhibit many of the features of the duality.  ...  At higher energy scales, this emergent bulk geometry is modified in a way that matches the toy models of black holes proposed previously for HQECC.  ...  Acknowledgements The authors would like to thank Eva Silverstein for very helpful discussions about holographic dualities, and Fernando Pastawski for explaining holographic tensor network constructions  ... 
arXiv:1810.08992v2 fatcat:sc2ubzcqzzb7tnoykwmlyx2fqy

A Family of Optimal Locally Recoverable Codes

Itzhak Tamo, Alexander Barg
2014 IEEE Transactions on Information Theory  
We present a family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality.  ...  We also construct codes with several disjoint recovering sets for every symbol.  ...  A related locality property, introduced in [3] and [8] , is called maximally recoverable codes.  ... 
doi:10.1109/tit.2014.2321280 fatcat:wdx3qwkmifet3dgdmtgxs2tuea

Toy models of holographic duality between local Hamiltonians

Tamara Kohler, Toby Cubitt
2019 Journal of High Energy Physics  
Holographic quantum error correcting codes (HQECC) have been proposed as toy models for the AdS/CFT correspondence, and exhibit many of the features of the duality.  ...  At higher energy scales, this emergent bulk geometry is modified in a way that matches the toy models of black holes proposed previously for HQECC.  ...  Acknowledgments The authors would like to thank Eva Silverstein for very helpful discussions about holographic dualities, and Fernando Pastawski for explaining holographic tensor network constructions,  ... 
doi:10.1007/jhep08(2019)017 fatcat:igsnydschbgu5c6aajcjwh5qsi

A family of optimal locally recoverable codes

Itzhak Tamo, Alexander Barg
2014 2014 IEEE International Symposium on Information Theory  
We present a family of LRC codes that attain the maximum possible value of the distance for a given locality parameter and code cardinality.  ...  We also construct codes with several disjoint recovering sets for every symbol.  ...  A related locality property, introduced in [3] and [8] , is called maximally recoverable codes.  ... 
doi:10.1109/isit.2014.6874920 dblp:conf/isit/TamoB14 fatcat:phfqvgepo5f7vhzpeausqrd4zu

Bulk entanglement gravity without a boundary: Towards finding Einstein's equation in Hilbert space

ChunJun Cao, Sean M. Carroll
2018 Physical Review D  
For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along codimension-one surfaces with the entanglement entropy between  ...  the full quantum-gravity Hilbert space and the semiclassical limit of quantum fields propagating on a classical spacetime.  ...  We also thank Aidan Chatwin-Davies, Junyu Liu, Spiros Michalakis, and Gunther Uhlmann for engaging and informing discussions.  ... 
doi:10.1103/physrevd.97.086003 fatcat:7wfwh5o26bd7nk7v4n4odazwla

Linear Programming Approximations for Index Coding [article]

Abhishek Agarwal, Larkin Flodin, Arya Mazumdar
2019 arXiv   pre-print
Index coding, a source coding problem over broadcast channels, has been a subject of both theoretical and practical interest since its introduction (by Birk and Kol, 1998).  ...  The graph G has a vertex v_i for each client and a directed edge (v_i, v_j) indicating that client i knows the jth symbol of the input.  ...  A part of the paper was presented in the Network Coding and Applications (NetCod 2016), IEEE GLOBECOM Workshop, Dec 2016 [1] .  ... 
arXiv:1807.07193v3 fatcat:n5vis34if5aerlstcrc6bpdmja

Effect of stress triaxiality on void growth in dynamic fracture of metals: A molecular dynamics study

E. T. Seppälä, J. Belak, R. E. Rudd
2004 Physical Review B  
Molecular dynamics (MD) simulations using an embedded-atom (EAM) potential for copper have been performed at room temperature and using strain controlling with high strain rates ranging from 10^7/sec to  ...  The stress thresholds for void growth are found to be comparable with spall strength values determined by dynamic fracture experiments.  ...  Richard Becker bringing to our attention Refs. 51 and 52 that find the prolate-oblate transition in continuum modeling.  ... 
doi:10.1103/physrevb.69.134101 fatcat:amse3vnw2jhdhgib6j73kcorza

Private Retrieval, Computing and Learning: Recent Progress and Future Challenges [article]

Sennur Ulukus, Salman Avestimehr, Michael Gastpar, Syed Jafar, Ravi Tandon, Chao Tian
2021 arXiv   pre-print
In addition, the cross-cutting techniques and interconnections between the three topics are discussed along with a set of open problems and challenges.  ...  The article motivates each privacy setting, describes the problem formulation, summarizes breakthrough results in the history of each problem, and gives recent results and discusses some of the major ideas  ...  C represents capacity, C ∞ is the asymptotic capacity for large number of messages (K → ∞), C u and C l are upper and lower bounds on C, respectively.  ... 
arXiv:2108.00026v1 fatcat:6guz4ejeuvfffi6e46abosf7vq

Hydrogen-bond relaxation dynamics: Resolving mysteries of water ice

Yongli Huang, Xi Zhang, Zengsheng Ma, Yichun Zhou, Weitao Zheng, Ji Zhou, Chang Q. Sun
2015 Coordination chemistry reviews  
, relegation and warm water cools faster. progress shows that O:H-O bond segmental disparity and O-O repulsivity form the soul dictating the extraordinary adaptivity, cooperativity, recoverability, and  ...  sensitivity of water and ice.  ...  calculated using the MD method to derive the stress tensors.  ... 
doi:10.1016/j.ccr.2014.10.003 fatcat:yyywtriv5jdnhc4k4yejcugw6i
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