  An $(a,b,\tau)$ streaming code is a packet-level erasure code that can
recover under a strict delay constraint of $\tau$ time units, from either a
burst of $b$ erasures or else of $a$ random erasures, occurring within a
sliding window of time duration $w$. While rate-optimal constructions of such
streaming codes are available for all parameters $\{a,b,\tau,w\}$ in the
literature, they require in most instances, a quadratic, $O(\tau^2)$ field
size. In this work, we make further progress towards field size reduction and
present rate-optimal $O(\tau)$ field size streaming codes for two regimes: (i)
$gcd(b,\tau+1-a)\ge a$ (ii) $\tau+1 \ge a+b$ and $b \mod \ a \in \{0,a-1\}$.
