On Streaming Codes for Simultaneously Correcting Burst and Random Erasures

Streaming codes are packet-level codes that recover dropped packets within a strict decoding-delay constraint. We study streaming codes over a sliding-window (SW) channel model which admits only those erasure patterns which allow either a single burst erasure of
$\leq b$
packets along with
$\leq e$
random packet erasures, or else,
$\leq a$
random packet erasures, in any sliding-window of
$w$
time slots. We determine the optimal rate of a streaming code constructed via the popular diagonal embedding (DE) technique over such a SW channel under delay constraint
$\tau=(w-1)$
and provide an
$O(w)$
field size code construction. For the case
$e > 1$
, we show that it is not possible to significantly reduce this field size requirement, assuming the well-known MDS conjecture. We then provide a block code construction whose DE yields a streaming code achieving the rate derived above, over a field of size sub-linear in
$w$
, for a family of parameters having
$e=1$
. We show the field size optimality of this construction for some parameters, and near-optimality for others under a sparsity constraint. Additionally, we derive an upper-bound on the minimum distance of a cyclic code and characterize cyclic codes which achieve this bound via their ability to simultaneously recover from burst and random erasures.