  In this paper, for any fixed positive integers $t$ and $q>2$, we construct
$q$-ary codes correcting a burst of at most $t$ deletions with redundancy $\log
n+8\log\log n+o(\log\log n)+\gamma_{q,t}$ bits and near-linear
encoding/decoding complexity, where $n$ is the message length and
$\gamma_{q,t}$ is a constant that only depends on $q$ and $t$. In previous
works there are constructions of such codes with redundancy $\log n+O(\log
q\log\log n)$ bits or $\log n+O(t^2\log\log n)+O(t\log q)$. The redundancy of
our new construction is independent of $q$ and $t$ in the second term.
