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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