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Lower Bounds for Maximally Recoverable Tensor Code and Higher Order MDS Codes
[article]
2021
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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)
2018
Dagstuhl Reports
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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]
2021
arXiv
pre-print
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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]
2020
arXiv
pre-print
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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]
2021
arXiv
pre-print
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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]
2019
arXiv
pre-print
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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
2014
IEEE Transactions on Information Theory
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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
2019
Journal of High Energy Physics
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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
2014
2014 IEEE International Symposium on Information Theory
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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
2018
Physical Review D
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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]
2019
arXiv
pre-print
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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
2004
Physical Review B
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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]
2021
arXiv
pre-print
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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
2015
Coordination chemistry reviews
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2018 pre-print arXiv:1809.01523v1 fatcat:w4wskoicnrfcjlxs4sqt57nwwu
, 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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