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Upper bounds on weight hierarchies of extremal non-chain codes

Hans Georg Schaathun
2001 Discrete Mathematics  
This paper gives upper bounds on the weight hierarchies of the latter class of codes.  ...  Linear codes may be classiÿed according to a set of chain and non-chain conditions, the extreme cases being codes satisfying the chain condition (due to Wei and Yang) and extremal, non-chain codes (due  ...  Acknowledgements The author wishes to thank prof. Torleiv KlHve for helpful comments and for suggesting the problem in the ÿrst place.  ... 
doi:10.1016/s0012-365x(01)00131-5 fatcat:jksglclvczgkxheuou7wbm66nm

Page 3403 of Mathematical Reviews Vol. , Issue 98E [page]

1998 Mathematical Reviews  
) Bounds on the weight hierarchies of extremal non-chain codes of dimension 4.  ...  Bounds on the weight hierarchies of such codes of dimension 4 are studied.” 98e:94028 94B05 Constantinescu, loana (D-MUTU; Munich); Heise, Werner (D-MUTU; Munich) On the concept of code-isomorphy.  ... 

Page 9502 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews  
Han (D-ESSN-EM; Essen) The determination of the chain good weight hierarchies with high dimension. (English summary) SIAM J. Discrete Math. 17 (2003), no. 2, 196-209 (electronic).  ...  known bounds on the minimum weight for linear codes over Fs; are constructed from conference matrices.  ... 

X-code: MDS array codes with optimal encoding

Lihao Xu, J. Bruck
1999 IEEE Transactions on Information Theory  
The X-codes are of minimum column distance 3, namely, they can correct either one column error or two column erasures.  ...  We present a new class of MDS array codes of size n 2 n (n a prime number) called X-code.  ...  In [5] we gave bounds on the weight hierarchies of extremal nonchain codes of dimension 4.  ... 
doi:10.1109/18.746809 fatcat:35s4hw72lrhrljbjrtqa2nbqpy

On Higher Weights and Code Existence [chapter]

Hans Georg Schaathun
2009 Lecture Notes in Computer Science  
In this paper we suggest one approach based on the weight hierarchy, and it allows us to conduct an extremely rapid computer search to prove that there are exactly two inequivalent [36, 8, 16] codes.  ...  The technique can also be used to gain new information about the weight hierarchy of the putative [72, 36, 16] code, but not yet enough to say if it exists or not.  ...  Every binary [36, 8, 16] code satisfies the chain condition. Proof. Consider a subcode D 4 30 of dimension 4 and weight 30.  ... 
doi:10.1007/978-3-642-10868-6_4 fatcat:2eycysad4vfcfp74mfwfmouzpq

Weight hierarchies of binary linear codes of dimension 4

Wende Chen, Torleiv Kløve
2001 Discrete Mathematics  
The codes of dimension 4 are collected in classes. The possible weight hierarchies in each class are given. For one class the details of the proofs are included.  ...  The weight hierarchy of a binary linear [n; k] code C is the sequence (d1; d2; : : : ; d k ) where dr is the smallest support of an r-dimensional subcode of C.  ...  The research was supported by The Norwegian Research Council and the National Science Foundation of China.  ... 
doi:10.1016/s0012-365x(00)00411-8 fatcat:xirrhto2mzdkreh27dw3wr7qau

Page 8142 of Mathematical Reviews Vol. , Issue 99k [page]

1999 Mathematical Reviews  
, Wende (PRC-ASBJ-SYC; Beijing) ; Kleve, Torleivy (N-BERG-I; Bergen) Weight hierarchies of extremal non-chain binary codes of dimension 4, (English summary) IEEE Trans.  ...  The possible weight hierarchies of such binary codes of dimension 4 are determined.” 99k:94046 94B05 Hamada, Noboru (J-OWU-AM; Osaka) On the nonexistence of some quaternary linear codes meeting the Griesmer  ... 

Quantum error correction with fractal topological codes [article]

Arpit Dua, Tomas Jochym-O'Connor, Guanyu Zhu
2022 arXiv   pre-print
Recently, a class of fractal surface codes (FSCs), has been constructed on fractal lattices with Hausdorff dimension 2+ϵ, which admits a fault-tolerant non-Clifford CCZ gate.  ...  The latter can be mapped to the critical point of a zero-temperature confinement-Higgs transition on the fractal lattice, which is tunable via the Hausdorff dimension.  ...  The code used for the simulations and the data extracted are currently available upon request. Both will be made publicly available on GitHub shortly.  ... 
arXiv:2201.03568v1 fatcat:edwdzgwwwzhptcevgh5k5qehcm

Author index

2001 Discrete Mathematics  
., Helge Tverberg A celebration of a life in mathematics (1}3) 1} 6 Schaathun , H.G., Upper bounds on weight hierarchies of extremal non-chain codes (1}3) 449}469 SchoK bel, A., see H.  ...  Kl+ve, On the second greedy weight for linear codes of dimension 3 (1}3) 171}187 Daneshgar, A., A.J.W. Hilton and P.D.  ... 
doi:10.1016/s0012-365x(01)00341-7 fatcat:waxxkib62rgktblf37dt666hiq

Quantum Pin Codes [article]

Christophe Vuillot, Nikolas P. Breuckmann
2022 arXiv   pre-print
We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin codes are a generalization of quantum color codes and Reed-Muller codes and share a lot of their structure and properties.  ...  We study examples of these codes and their properties.  ...  One can also prove the following general lower bound on the distance of pin codes. Proposition 7 (Distance at least 4).  ... 
arXiv:1906.11394v3 fatcat:bk3djrn5g5ayrjaojgoixeo72y

Lower bound for the cost of connecting tree with given vertex degree sequence [article]

Mikhail Goubko, Alexander Kuznetsov
2018 arXiv   pre-print
The proposed lower bound estimate is used to construct several heuristic algorithms and to evaluate their quality on a variety of generated and real-life data sets.  ...  For the case of connecting trees with the given sequence of vertex degrees, the cost of the optimal tree is shown to be bounded from below by the solution of a semidefinite optimization program with bilinear  ...  The quality of the lower bound is evaluated on a number of generated flow matrices with dimension from 10 to 1000 and on the selected real-life origindestination matrices with dimension varying from 12  ... 
arXiv:1808.06199v2 fatcat:v5l2zfmgrfa3th7vlwzbgthguu

Evolutionary Trends of Developer Coordination: A Network Approach [article]

Mitchell Joblin, Sven Apel, Wolfgang Mauerer
2016 arXiv   pre-print
to achieve this state depend on the project's scale.  ...  In particular, we found that: (a) developers form scale-free networks, in which the majority of coordination requirements arise among an extremely small number of developers, (b) developers tend to accumulate  ...  Acknowledgements This work has been supported by the German Research Foundation (AP 206/4, AP 206/5, AP 206/6).  ... 
arXiv:1510.06988v2 fatcat:uizeghe33jevtehg4ebmrgkhgq

Decoupling algorithms from schedules for easy optimization of image processing pipelines

Jonathan Ragan-Kelley, Andrew Adams, Sylvain Paris, Marc Levoy, Saman Amarasinghe, Frédo Durand
2012 ACM Transactions on Graphics  
code.  ...  Unfortunately, optimization usually comes at the cost of programmer pain and code complexity, as computation must be reorganized to achieve memory efficiency and parallelism.  ...  Acknowledgments This work was partially funded by the Quanta T-Party, NSF grants 0964004, 0964218, and 0832997, DOE award DE-SC0005288, and gifts from Cognex and Adobe.  ... 
doi:10.1145/2185520.2185528 fatcat:mdfbhjwc5zatfoq4mt6p3j7m7a

Decoupling algorithms from schedules for easy optimization of image processing pipelines

Jonathan Ragan-Kelley, Andrew Adams, Sylvain Paris, Marc Levoy, Saman Amarasinghe, Frédo Durand
2012 ACM Transactions on Graphics  
code.  ...  Unfortunately, optimization usually comes at the cost of programmer pain and code complexity, as computation must be reorganized to achieve memory efficiency and parallelism.  ...  Acknowledgments This work was partially funded by the Quanta T-Party, NSF grants 0964004, 0964218, and 0832997, DOE award DE-SC0005288, and gifts from Cognex and Adobe.  ... 
doi:10.1145/2185520.2335383 fatcat:smf4xdhjpzhn3ggl2qyilol22y

On the trellis structure of block codes

F.R. Kschischang, V. Sorokine
1995 IEEE Transactions on Information Theory  
Minimal trellises for linear codes are obtained by forming the product of elementary trellises corresponding to the one-dimensional subcodes generated by atomic codewords.  ...  A correspondence between minimal linear block code trellises and con gurations of non-attacking rooks on a triangular chess board is established and used to show that the number of distinct minimal linear  ...  Trott for prompting us to discover the example described in the Appendix.  ... 
doi:10.1109/18.476317 fatcat:qmjwknh4rzfgvjedigcwo2jowu
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