## Optimal frames for erasures (2004)

Venue: | Linear Algebra Appl |

Citations: | 36 - 4 self |

### BibTeX

@ARTICLE{Holmes04optimalframes,

author = {Roderick B. Holmes and Vern and I. Paulsen},

title = {Optimal frames for erasures},

journal = {Linear Algebra Appl},

year = {2004},

pages = {31--51}

}

### OpenURL

### Abstract

Abstract. We study frames from the viewpoint of coding theory. We introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting. 1.

### Citations

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- 2003
(Show Context)
Citation Context ...t, when they exist, we prove are optimal. After writing a preliminary draft of this paper we learned that this family of frames was also being studied independently by Thomas Strohmer and Robert Heath=-=[25]-=- and we have incorporated a number of their observations into this paper. We begin by recalling the basic definitions and concepts. Let H be a Hilbert space, real or complex, and let F = {fi}i∈I ⊂ H b... |

144 | Quantized frame expansions with erasures
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Citation Context ...w well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting. 1. Introduction In [8], =-=[16]-=- and [17] the family of uniform tight frames are studied from a coding theory viewpoint and these frames are shown to be optimal in some sense for one erasure. They then develop further properties of ... |

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Citation Context ...frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting. 1. Introduction In [8], [16] and =-=[17]-=- the family of uniform tight frames are studied from a coding theory viewpoint and these frames are shown to be optimal in some sense for one erasure. They then develop further properties of these fra... |

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12 |
The Pythagorean theorem I: the finite case
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(Show Context)
Citation Context ... also an isometry. By choosing θ appropriately, we can insure that ‖gi‖2 = k/n. Repeating this process at most n − 1 times we obtain a uniform (n, k)-frame. This algorithm is essentially adopted from =-=[22]-=-. There is another place in the literature where uniform (n, k)-frames arise, but in a different guise. A finite subset of vectors {x1, . . . , xn} on the unitOPTIMAL FRAMES FOR ERASURES 5 sphere Sk−... |

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Symmetric Hadamard matrices of order 36
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Citation Context ...eans, one finds that necessarily n, 8k + 1 and 8(n − k) + 1 need to be perfect squares. We have seen that examples exist for n = 4 j . The next smallest possible value is n = 36. Bussemaker and Seidel=-=[3]-=- show that there are 92 symmetric Hadamard matrices of this size and so these matrices yield real 2-uniform (36, k)-frames for k = 15, 21. In algebra, two Hadamard matrices H1 and H2 are usually consi... |

9 |
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Citation Context ...I + 6Q is exhibited and thus one obtains a 2-uniform (28, 7)-frame. This signature matrix is obtained from the adjacency matrix of the first of the strongly regular graphs on 28 vertices appearing in =-=[24]-=- by replacing its’ standard adjacency matrix by its’ Seidel adjacency matrix. Given a graph G on n vertices, the Seidel adjacency matrix of G is defined to be the n × n matrix A = (ai,j) where ai,j is... |

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Citation Context ...nd the optimal packing that they describe yields a 2-uniform (4, 3)-frame. Example 3.6. Conference Matrices. A real n × n matrix C with ci,i = 0 and ci,j = ±1 for i ̸= j is called a conference matrix =-=[13]-=- provided C ∗ C = (n − 1)I.OPTIMAL FRAMES FOR ERASURES 19 Thus, every symmetric conference matrix is a signature matrix with µ = 0 and k = n/2. So, in particular such matrices must be of even size an... |

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2 |
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(Show Context)
Citation Context ...s a uniform (n, 3)-frame. If one does obtain a uniform (n, 3)-frame from one of their packings, then it is necessarily a frame in E2(n, 3) and in such a case their angle and Θn,3 will be equal. Fickus=-=[15]-=- shows that for various uniform solids, by choosing the unit vectors corresponding to the vertices, one obtains a tight frame. When these solids are symmetric under antipodal reflection, then one also... |

2 |
A survey of two-graphs,Proc. Intern
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Citation Context ... an even number of the sets from ∆. A two-graph is regular, provided that every two element subset of Ω is contained in the same number, α, of sets in ∆.OPTIMAL FRAMES FOR ERASURES 23 Given n, Seidel=-=[23]-=- exhibits a one-to-one correspondence between the twographs on the set of n elements and the switching equivalence classes of graphs on n elements and gives a concrete means, given the two-graph, to c... |

1 | The art of frame theory, preprint, math.FA/9910168 at xxx.lanl.gov - Casazza - 1999 |

1 |
Uniform tight frames with erasures, preprint
- Casazza, Kovacevic
(Show Context)
Citation Context ...of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting. 1. Introduction In =-=[8]-=-, [16] and [17] the family of uniform tight frames are studied from a coding theory viewpoint and these frames are shown to be optimal in some sense for one erasure. They then develop further properti... |