Binary frames, graphs and erasures

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.