  Synchronization strings are recently introduced by Haeupler and Shahrasbi
(STOC 2017) in the study of codes for correcting insertion and deletion errors
(insdel codes). They showed that for any parameter $\varepsilon>0$,
synchronization strings of arbitrary length exist over an alphabet whose size
depends only on $\varepsilon$. Specifically, they obtained an alphabet size of
$O(\varepsilon^{-4})$, which left an open question on where the minimal size of
such alphabets lies between $\Omega(\varepsilon^{-1})$ and
$O(\varepsilon^{-4})$. In this work, we partially bridge this gap by providing
an improved lower bound of $\Omega(\varepsilon^{-3/2})$, and an improved upper
bound of $O(\varepsilon^{-2})$. We also provide fast explicit constructions of
synchronization strings over small alphabets.
  Further, along the lines of previous work on similar combinatorial objects,
we study the extremal question of the smallest possible alphabet size over
which synchronization strings can exist for some constant $\varepsilon < 1$. We
show that one can construct $\varepsilon$-synchronization strings over
alphabets of size four while no such string exists over binary alphabets. This
reduces the extremal question to whether synchronization strings exist over
ternary alphabets.
