  It is well known that a random q-ary code of rate \Omega(\epsilon^2) is list
decodable up to radius (1 - 1/q - \epsilon) with list sizes on the order of
1/\epsilon^2, with probability 1 - o(1). However, until recently, a similar
statement about random linear codes has until remained elusive. In a recent
paper, Cheraghchi, Guruswami, and Velingker show a connection between list
decodability of random linear codes and the Restricted Isometry Property from
compressed sensing, and use this connection to prove that a random linear code
of rate \Omega(\epsilon^2 / log^3(1/\epsilon)) achieves the list decoding
properties above, with constant probability. We improve on their result to show
that in fact we may take the rate to be \Omega(\epsilon^2), which is optimal,
and further that the success probability is 1 - o(1), rather than constant. As
an added benefit, our proof is relatively simple. Finally, we extend our
methods to more general ensembles of linear codes. As an example, we show that
randomly punctured Reed-Muller codes have the same list decoding properties as
the original codes, even when the rate is improved to a constant.
