  Data erasure can often occur in communication. Guarding against erasures
involves redundancy in data representation. Mathematically this may be achieved
by redundancy through the use of frames. One way to measure the robustness of a
frame against erasures is to examine the worst case condition number of the
frame with a certain number of vectors erased from the frame. The term {\em
numerically erasure-robust frames (NERFs)} was introduced in \cite{FicMix12} to
give a more precise characterization of erasure robustness of frames. In the
paper the authors established that random frames whose entries are drawn
independently from the standard normal distribution can be robust against up to
approximately 15\% erasures, and asked whether there exist frames that are
robust against erasures of more than 50\%. In this paper we show that with very
high probability random frames are, independent of the dimension, robust
against any amount of erasures as long as the number of remaining vectors is at
least $1+\delta$ times the dimension for some $\delta_0>0$. This is the best
possible result, and it also implies that the proportion of erasures can
arbitrarily close to 1 while still maintaining robustness. Our result depends
crucially on a new estimate for the smallest singular value of a rectangular
random matrix with independent standard normal entries.
