  In this paper, we design erasure-correcting codes for channels with burst and
random erasures, when a strict decoding delay constraint is in place. We
consider the sliding-window-based packet erasure model proposed by Badr et al.,
where any time-window of width $w$ contains either up to $a$ random erasures or
an erasure burst of length at most $b$. One needs to recover any erased packet,
where erasures are as per the channel model, with a strict decoding delay
deadline of $\tau$ time slots. Presently existing rate-optimal constructions in
the literature require, in general, a field-size which grows exponential in
$\tau$, for a constant $\frac{a}{\tau}$. In this work, we present a new
rate-optimal code construction covering all channel and delay parameters, which
requires an $O(\tau^2)$ field-size. As a special case, when $(b-a)=1$, we have
a field-size linear in $\tau$. We also present three other constructions having
linear field-size, under certain constraints on channel and decoding delay
parameters. As a corollary, we obtain low field-size, rate-optimal
convolutional codes for any given column distance and column span. Simulations
indicate that the newly proposed streaming code constructions offer lower
packet-loss probabilities compared to existing schemes, for selected instances
of Gilbert-Elliott and Fritchman channels.
