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 $b mod a in {0,a-1}$.
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