  We demonstrate that a high fidelity approximation to $| \Psi_b \rangle$, the
quantum superposition over all bit strings within Hamming distance $b$ of the
codewords of a dimension-$k$ linear code over $\mathbb{Z}_2^n$, can be
efficiently constructed by a quantum circuit for large values of $n$, $b$ and
$k$ which we characterize. We do numerical experiments at $n=1000$ which back
up our claims. The achievable radius $b$ is much larger than the distance out
to which known classical algorithms can efficiently find the nearest codeword.
Hence, these states cannot be prepared by quantum constuctions that require
uncomputing to find the codeword nearest a string. Unlike the analogous states
for lattices in $\mathbb{R}^n$, $|\Psi_b \rangle$ is not a useful resource for
bounded distance decoding because the relevant overlap falls off too quickly
with distance and known classical algorithms do better. Furthermore the overlap
calculation can be dequantized. Perhaps these states could be used to solve
other code problems. The technique used to construct these states is of
interest and hopefully will have applications beyond codes.
