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Maximal Arcs in Projective Three-Spaces and Double-Error-Correcting Cyclic Codes

Henk D.L. Hollmann, Qing Xiang
2001 Journal of combinatorial theory. Series A  
Using maximal arcs in PG(3, 2 m ), we give a new proof of the fact that the binary cyclic code C (m) 1, 2 2h &2 h +1 , the code of length 2 m &1 with defining zeroes : and : t , t=2 2h &2 h +1, where :  ...  is a primitive element in GF(2 m ), is 2-error-correcting when gcd(m, h)=1.  ...  In this note we give a short new proof by using maximal arcs in PG(3, 2 m ), the projective 3-space over GF(2 m ). THE NEW PROOF We first state the result we want to prove. Theorem 1.  ... 
doi:10.1006/jcta.2000.3064 fatcat:q2cee5tgorgcbhfhctgwptxone

Page 8555 of Mathematical Reviews Vol. , Issue 2001K [page]

2001 Mathematical Reviews  
(NL-PRL; Eindhoven); Xiang, Qing (1-DE; Newark, DE) Maximal arcs in projective three-spaces and double-error-correcting cyclic codes. (English summary) J. Combin. Theory Ser.  ...  The subgroup can be used to obtain a y(m)-dimensional signal space and to construct block codes, over algebraic inte- gers, which are able to correct some error patterns, where ¢ is the Euler function.  ... 

New lower bounds on the size of (n,r)-arcs in PG(2,q) [article]

Michael Braun
2019 arXiv   pre-print
An (n,r)-arc in PG(2,q) is a set of n points such that each line contains at most r of the selected points. It is well-known that (n,r)-arcs in PG(2,q) correspond to projective linear codes.  ...  Let m_r(2,q) denote the maximal number n of points for which an (n,r)-arc in PG(2,q) exists. In this paper we obtain improved lower bounds on m_r(2,q) by explicitly constructing (n,r)-arcs.  ...  Ball and A. Blokhuis, On the size of a double blocking set in PG(2, q). Finite Fields Appl., 2 (1996) 125-137. [2] S. Ball, Three-dimensional linear codes, online table,  ... 
arXiv:1808.02702v2 fatcat:hwksrn6w35gdplp73ioy63544e

M-theory, Black Holes and Cosmology [article]

Renata Kallosh
2020 arXiv   pre-print
Octonions and Hamming error correcting codes are at the base of these models. They lead to 7 benchmark targets of future CMB missions looking for primordial gravitational wave from inflation.  ...  I also show puzzling relations between the fermion mass eigenvalues in these cosmological models, exceptional Jordan eigenvalue problem, and black hole entropy.  ...  Then a projective geometry P (V ) can be 7.2 Cyclic codes and projective spaces As shown in Sect. 2.3, a linear code C is a subspace of F n q , q = p m .  ... 
arXiv:2009.11339v1 fatcat:626w5j57o5bgxchnsk2saau47q

Moderate Density Parity-Check Codes from Projective Bundles [article]

Jessica Bariffi, Sam Mattheus, Alessandro Neri, Joachim Rosenthal
2021 arXiv   pre-print
In this setting, our codes have the best possible error-correction performance for this range of parameters.  ...  We determine minimum distance and dimension of these codes, showing that they have a natural quasi-cyclic structure.  ...  MDPC codes from Projective Planes The projective plane PG(2, q) is a point-line geometry constructed from a three-dimensional vector space V over F q .  ... 
arXiv:2103.09722v1 fatcat:tsvuul4l3zedfggz5klvs6pj2m

Finite geometries

A. Blokhuis, J. W. P. Hirschfeld, D. Jungnickel, J. A. Thas
2008 Designs, Codes and Cryptography  
Landjev, On arcs in projective Hjelmslev planes over finite chain rings D. Leemans, RWPRI and (2T ) 1 flag-transitive linear spaces G. Lunardon, Spreads in H(q) and 1-systems of Q(6, q) D.  ...  Hamilton, New constructions of maximal arcs in Desarguesian projective planes J. Jedwab, Designing the IEEE 802.12 transmission code L.H. Khachatrian, Extremal problems under dimension constraints G.  ...  A class of designs protecting against quantum jumps Thomas Beth Quantum Error-Correcting Codes and their intrinsic relation to Self-Dual Codes and Finite Geometries have been known as a hot topic of  ... 
doi:10.1007/s10623-007-9162-6 fatcat:yd5cqfmlrfa7zlzeifhhx7qfg4

A cyclical deep learning based framework for simultaneous inverse and forward design of nanophotonic metasurfaces

Abhishek Mall, Abhijeet Patil, Amit Sethi, Anshuman Kumar
2020 Scientific Reports  
Importantly, our cyclical generation framework also explores the space of new metasurface topologies.  ...  This is a highly iterative process based on trial and error, which is computationally costly and time consuming.  ...  The MSE threshold defined on the pGA was 0.037 which is the sum of the SNN and cGAN average MSE considering the cumulative propagation of errors double-arc (c) rectangle-circle pair (d) rectangle-square  ... 
doi:10.1038/s41598-020-76400-y pmid:33173073 fatcat:ju7vl2cmuje6re5ocy7zb4avoy

Exordium for DNA Codes

Arkadii G. D'yachkov, Peter L. Erdös, Anthony J. Macula, Vyacheslav V. Rykov, David C. Torney, Chang-Shung Tung, Pavel A. Vilenkin, P. Scott White
2003 Journal of combinatorial optimization  
We describe how deletion-correcting codes may be enhanced to yield codes with double-strand DNAsequence codewords.  ...  We mention motivating applications of DNA codes.  ...  Bell and Walter B. Goad, pioneers in bioinformatics and founders of the Theoretical Biology Group at Los Alamos National Laboratory.  ... 
doi:10.1023/b:joco.0000017385.39168.0d fatcat:ut76l22z4bdwvp6oo7jqybokte

The number of points on a curve, and applications Arcs and curves: the legacy of Beniamino Segre

J. W. P. Hirschfeld
2006 Rendiconti di Matematica e delle sue Applicazioni. Serie VII  
Curves defined over a finite field have various applications, such as (a) the construction of good error-correcting codes, (b) the correspondence with arcs in a finite Desarguesian plane, (c) the Main  ...  Various of these results and their applications are surveyed.  ...  (PROJECTIVE SPACE) an n-arc in PG(k − 1, q), that is, a set K of n points with at most k − 1 in any hyperplane of the projective space of k − 1 dimensions over F q .  ... 
doaj:66a44ea000e4464d9c61fd5c205b20fe fatcat:y6ykm7h46nfkdnk5r4qssrg6gi

Network Coding Theory: A Survey

Riccardo Bassoli, Hugo Marques, Jonathan Rodriguez, Kenneth W. Shum, Rahim Tafazolli
2013 IEEE Communications Surveys and Tutorials  
.: NETWORK CODING THEORY: A SURVEY 3 by malicious nodes. The main characteristic of the network extension of error correction is that redundancy is in space domain instead of in time domain.  ...  Next, in some scenarios, scalar linear network coding is not enough to solve the network and non-linear network coding is necessary; actually, vector network coding represents a way to achieve solvability  ...  In particular, in the last scenario, the algorithms used tools from arcs in projective spaces to handle coding vectors.  ... 
doi:10.1109/surv.2013.013013.00104 fatcat:jgnf4i7rczfqrcyv7tygrydtxm

Subject Index Volumes 1–200

2001 Discrete Mathematics  
Erdös and Sós, 4925 Erdös and Szekeres, 4879 Erdös et al., 5529, 5888 Erdös Rubin Taylor, 3815 Erdös Sós Turán, 506 Everett, 2901 Faigle and Sands, 2122 Faudree et al., 4399 Faudree Gould Jacobson  ...  Slater, 5299 Hertz and de Werra, 3748 Hibi, 4839 Hoàng, 3748, 4811 Holton, 1660 Holyer and Cockayne, 1118 Howorka, 2636  ...  , see also two-error 2-error correcting BCH code, 1899 code, 1697 strongly uniformly packed code, 2985 2-error detecting, 1894 2-error-correcting, see 2-error correcting 2-error-detecting, see 2-error  ... 
doi:10.1016/s0012-365x(00)00471-4 fatcat:znxg54q6ifgutnhzj3urcw455y

The practitioner's guide to coloured Petri nets

Lars M. Kristensen, Soren Christensen, Kurt Jensen
1998 International Journal on Software Tools for Technology Transfer (STTT)  
CP-nets have a wide range of application areas and many CPN projects have been carried out in industry, e.g., in the areas of communication protocols, operating systems, hardware designs, embedded systems  ...  The tool is used by more than four hundred organisations in forty different countries -including one hundred commercial companies. It is available free of charge, also for commercial use.  ...  Many students and colleagues -in particular at the University of Aarhus and Meta Software -have influenced the development of CP-nets, their analysis methods, and their tool support.  ... 
doi:10.1007/s100090050021 fatcat:tp4gsxf7hvf7zpm5rih2fh4mda

A complete framework for controller verification in manufacturing

Christian Gerber, Sebastian Preusse, Hans-Michael Hanisch
2010 2010 IEEE 15th Conference on Emerging Technologies & Factory Automation (ETFA 2010)  
The contribution therefore proposes an approach to generate formal models out of PLC code. These controller models enable formal verification of the closed loop in combination with a specification.  ...  The results of analysis of open-loop controller behavior give very little or almost no indications of the correct behavior of the closed-loop system.  ...  At best, they can be corrected before leading to dangerous situations, but often, rare and critical errors are overlooked and can lead to a sudden plant breakdown or unexpected behavior.  ... 
doi:10.1109/etfa.2010.5641220 dblp:conf/etfa/GerberPH10 fatcat:x2kjuzsnq5ffno4pxtzydkgn34

Homological error correction: Classical and quantum codes

H. Bombin, M. A. Martin-Delgado
2007 Journal of Mathematical Physics  
We prove several theorems characterizing the existence of homological error correction codes both classically and quantumly.  ...  In the particular case of toric codes we construct an optimal family and give an explicit proof of its optimality.  ...  In this case we obtain a pseudo-classical code, capable of correcting errors of the form σ x0 whenever x is correctable in the corresponding classical code.  ... 
doi:10.1063/1.2731356 fatcat:kfq2cvjzdvfj3jlwiteagnolpe

Complex character positioning based on a compatible flow model of multiple supports

R. Boulic, R. Mas-Sanso, D. Thalmann
1997 IEEE Transactions on Visualization and Computer Graphics  
By construction, Inverse Kinetics presents the same control architecture as Inverse Kinematics, and, thus, it shows equivalent computing cost and similar intuitive concepts.  ...  Furthermore, Inverse Kinetics for the center of mass and Inverse Kinematics for fixed end effectors can be combined to generate a posture displaying static balance, goal-oriented features, and an additional  ...  ACKNOWLEDGMENTS The authors wish to thank Jianhua Shen for the human deformation software and Zhiyong Huang for the TRACK environment providing the images with human deformation.  ... 
doi:10.1109/2945.620491 fatcat:rbiuyxngrfe43n7g2j3ijowcuu
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