  We revisit the fundamental problem of dictionary look-up with mismatches.
Given a set (dictionary) of $d$ strings of length $m$ and an integer $k$, we
must preprocess it into a data structure to answer the following queries: Given
a query string $Q$ of length $m$, find all strings in the dictionary that are
at Hamming distance at most $k$ from $Q$. Chan and Lewenstein (CPM 2015) showed
a data structure for $k = 1$ with optimal query time $O(m/w + occ)$, where $w$
is the size of a machine word and $occ$ is the size of the output. The data
structure occupies $O(w d \log^{1+\varepsilon} d)$ extra bits of space (beyond
the entropy-bounded space required to store the dictionary strings). In this
work we give a solution with similar bounds for a much wider range of values
$k$. Namely, we give a data structure that has $O(m/w + \log^k d + occ)$ query
time and uses $O(w d \log^k d)$ extra bits of space.
