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Binary frames,
graphs and erasures
Bernhard G. Bodmann, Bijan Camp and Dax Mahoney
Vol. 7 (2014), No. 2, 151–169
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Abstract |
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This paper examines binary codes from a frame-theoretic viewpoint. Binary Parseval frames have convenient encoding and decoding maps. We characterize binary Parseval frames that are robust to one or two erasures. These characterizations are given in terms of the associated Gram matrix and with graph-theoretic conditions. We illustrate these results with frames in lowest dimensions that are robust to one or two erasures. In addition, we present necessary conditions for correcting a larger number of erasures. As in a previous paper, we emphasize in which ways the binary theory differs from the theory of frames for real and complex Hilbert spaces. |
Keywords
frames, Parseval frames, finite-dimensional vector spaces,
binary numbers, codes, switching equivalence, Gram
matrices, adjacency matrix, graphs
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Mathematical Subject Classification 2010
Primary: 42C15
Secondary: 94B05, 05C50
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Milestones
Received: 25 June 2012
Revised: 14 September 2012
Accepted: 14 September 2012
Published: 16 November 2013
Communicated by Stephan Garcia
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Authors |
| Bernhard G. Bodmann | |
| Department of Mathematics University of Houston Houston, TX 77204 United States |
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| Bijan Camp | |
| Department of Psychology University of Minnesota Minneapolis, MN 55414 United States |
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| Dax Mahoney | |
| Department of Mathematics University of Houston Houston, TX 77204 United States |
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