  We consider $(n,k,l)$ MDS codes of length $n$, dimension $k$, and
subpacketization $l$ over a finite field $\mathbb{F}$. A codeword of such a
code consists of $n$ column-vectors of length $l$ over $\mathbb{F}$, with the
property that any $k$ of them suffice to recover the entire codeword. Each of
these $n$ vectors may be stored on a separate node in a network. If one of the
$n$ nodes fails, we can recover its content by downloading symbols from the
surviving nodes, and the total number of symbols downloaded in the worst case
is called the repair bandwidth of the code. By the cut-set bound, the repair
bandwidth of an $(n,k,l)$ MDS code is at least $l(n{-}1)/(n{-}k)$. There are
several constructions of MDS codes whose repair bandwidth meets or
asymptotically meets the cut-set bound. For example, Ye and Barg constructed
$(n,k,r^{n})$ Reed--Solomon codes that asymptotically meet the cut-set bound,
where $r = n-k$. Ye and Barg also constructed optimal-bandwidth and
optimal-update $(n,k,r^{n})$ MDS codes. Wang, Tamo, and Bruck constructed
optimal-bandwidth $(n, k, r^{n/(r+1)})$ MDS codes, and these codes have the
smallest known subpacketization for optimal-bandwidth MDS codes.
  A key idea in all these constructions is to represent certain integers in
base $r$. We show how this technique can be refined to improve the
subpacketization of the two MDS code constructions by Ye and Barg, while
achieving asymptotically optimal repair bandwidth. Specifically, when $r=s^{m}$
for an integer $s$,we obtain an $(n,k,s^{m+n-1})$ Reed--Solomon code and an
optimal-update $(n,k,s^{m+n-1})$ MDS code, both having asymptotically optimal
repair bandwidth. We also present an extension of this idea to reduce the
subpacketization of the Wang--Tamo--Bruck construction while achieving a
repair-by-transfer scheme with asymptotically optimal repair bandwidth.
