  Reed-Solomon codes have found many applications in practical storage systems,
but were until recently considered unsuitable for distributed storage
applications due to the widely-held belief that they have poor repair
bandwidth. The work of Guruswami and Wootters (STOC'16) has shown that one can
actually perform bandwidth-efficient linear repair with Reed-Solomon codes:
When the codes are over the field $\mathbb{F}_{q^t}$ and the number of parities
$r \geq q^s$, where $(t-s)$ divides $t$, there exists a linear scheme that
achieves a repair bandwidth of $(n-1)(t-s)\log_2 q$ bits. We extend this result
by showing the existence of such a linear repair scheme for every $1 \leq s <
t$. Moreover, our new schemes are optimal among all linear repair schemes for
Reed-Solomon codes when $n = q^t$ and $r = q^s$. Additionally, we improve the
lower bound on the repair bandwidth for Reed-Solomon codes, also established in
the work of Guruswami and Wootters.
