Vol. 7, No. 2, 2014

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ISSN: 1944-4184 (e-only)
ISSN: 1944-4176 (print)
Binary frames, graphs and erasures

Bernhard G. Bodmann, Bijan Camp and Dax Mahoney

Vol. 7 (2014), No. 2, 151–169
Abstract

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
Mathematical Subject Classification 2010
Primary: 42C15
Secondary: 94B05, 05C50
Milestones
Received: 25 June 2012
Revised: 14 September 2012
Accepted: 14 September 2012
Published: 16 November 2013

Communicated by Stephan Garcia
Authors
Bernhard G. Bodmann
Department of Mathematics
University of Houston
Houston, TX 77204
United States
Bijan Camp
Department of Psychology
University of Minnesota
Minneapolis, MN 55414
United States
Dax Mahoney
Department of Mathematics
University of Houston
Houston, TX 77204
United States