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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Bibliographic Details
Published in:Linear algebra and its applications Vol. 404; pp. 118 - 146
Main Authors: Bodmann, Bernhard G., Paulsen, Vern I.
Format: Journal Article
Language:English
Published: New York, NY Elsevier Inc 15.07.2005
Elsevier Science
Subjects:
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
Online Access:Get full text
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Summary: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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2005.02.016