Covering Radius and the Restricted Isometry Property

The Restricted Isometry Property or RIP introduced by Candes and Tao requires an n x p dictionary to act as a near isometry on all k-sparse signals. This paper provides a very simple condition under which a dictionary Phi(C) obtained by exponentiating codewords from a binary linear code C satisfies the RIP with high probability. The method is to bound the difference between the dictionary Phi(C) and a second dictionary Delta generated by a random Bernoulli process which is known to satisfy the RIP with high probability. The difference Delta - Phi(C) is controlled by the covering radius of C, a fundamental parameter that is bounded above by the number of weights in the dual code C-perpendicular to (the external distance of C). The main result complements a more sophisticated asymptotic analysis by Babadi and Tarokh of the distribution of eigenvalues of random submatrices of Phi(C). In this analysis, divergence from the distribution corresponding to the full Bernoulli matrix depends on a different fundamental parameter of C, namely the minimum distance of the dual code C-perpendicular to.