An (a, b, τ) streaming code is a packet-level erasure code that can recover under a strict delay constraint of τ 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, τ, w} in the literature, they require in most instances, a quadratic, O(τ2) field size. In this work, we make further progress towards field size reduction and present rate-optimal O(τ) field size streaming codes for two regimes: (i) gcd(b, τ + 1 - a) ≥ a (ii) τ + 1 a + b and b mod a 0, a - 1. © 2020 IEEE.