Frames, graphs and erasures

Two-uniform frames and their use for the coding of vectors are the main subject of this paper. These frames are known to be optimal for handling up to two erasures, in the sense that they minimize the largest possible error when up to two frame coefficients are set to zero. Here, we consider various...

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Vydané v:	Linear algebra and its applications Ročník 404; s. 118 - 146
Hlavní autori:	Bodmann, Bernhard G., Paulsen, Vern I.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York, NY Elsevier Inc 15.07.2005 Elsevier Science
Predmet:	Algebra Codes Conference matrix Erasures Error bounds Exact sciences and technology Frames Graphs Hadamard matrix Linear and multilinear algebra, matrix theory Mathematics Sciences and techniques of general use Two-graphs Frames Codes Erasures Primary 46L05 47A20 Hadamard matrix Secondary 46A22 Conference matrix Error bounds Graphs 46M10 Two-graphs 46H25 Error estimation 47A20 Frames Error bound Erasure Graph theory Code Equivalence classes Coding Primary Hadamard transformation
ISSN:	0024-3795, 1873-1856
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Abstract	Two-uniform frames and their use for the coding of vectors are the main subject of this paper. These frames are known to be optimal for handling up to two erasures, in the sense that they minimize the largest possible error when up to two frame coefficients are set to zero. Here, we consider various numerical measures for the reconstruction error associated with a frame when an arbitrary number of the frame coefficients of a vector are lost. We derive general error bounds for two-uniform frames when more than two erasures occur and apply these to concrete examples. We show that among the 227 known equivalence classes of two-uniform (36, 15)-frames arising from Hadamard matrices, there are 5 that give smallest error bounds for up to 8 erasures.
AbstractList	Two-uniform frames and their use for the coding of vectors are the main subject of this paper. These frames are known to be optimal for handling up to two erasures, in the sense that they minimize the largest possible error when up to two frame coefficients are set to zero. Here, we consider various numerical measures for the reconstruction error associated with a frame when an arbitrary number of the frame coefficients of a vector are lost. We derive general error bounds for two-uniform frames when more than two erasures occur and apply these to concrete examples. We show that among the 227 known equivalence classes of two-uniform (36, 15)-frames arising from Hadamard matrices, there are 5 that give smallest error bounds for up to 8 erasures.
Author	Bodmann, Bernhard G. Paulsen, Vern I.
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Cites_doi	10.1002/sapm1933121311 10.4153/CJM-1967-091-8 10.1023/A:1021349819855 10.1006/acha.2000.0340 10.1016/S1063-5203(03)00023-X 10.1007/978-1-4757-6568-7 10.1016/0012-365X(75)90029-1 10.1215/S0012-7094-44-01108-7 10.1007/BFb0092256 10.1016/j.laa.2003.07.012 10.1109/18.650985
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Keywords	Frames Codes Erasures Primary 46L05 47A20 Hadamard matrix Secondary 46A22 Conference matrix Error bounds Graphs 46M10 Two-graphs 46H25 Error estimation 47A20 Frames Error bound Erasure Graph theory Code Equivalence classes Coding Primary Hadamard transformation
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Title	Frames, graphs and erasures
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