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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 $\le b$ packets along with $\le e$ random packet erasures, or else, $\le a$ 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 $d_{\text{min}}$ of a cyclic code and characterize cyclic codes which achieve this bound via their ability to simultaneously recover from burst and random erasures.

Streaming codes (SCs) are packet-level codes that recover erased packets within a strict decoding-delay deadline. Streaming codes for various packet erasure channel models such as sliding-window (SW) channel models that admit random or burst erasures in any SW of a fixed length have been studied in the literature, and the optimal rate as well as rate-optimal code constructions of SCs over such channel models are known. In this paper, we study error-correcting streaming codes ($\text{SC}_{\text{ERR}}$s), i.e., packet-level codes which recover erroneous packets within a delay constraint. We study $\text{SC}_{\text{ERR}}$s for two classes of SW channel models, one that admits random packet errors, and another that admits multiple bursts of packet errors, in any SW of a fixed length. For the case of random packet errors, we establish the equivalence of an $\text{SC}_{\text{ERR}}$ and a corresponding SC that recovers from random packet erasures, thus determining the optimal rate of an $\text{SC}_{\text{ERR}}$ for this setting, and providing a rate-optimal code construction for all parameters. We then focus on SCs that recover from multiple erasure bursts and derive a rate-upper-bound for such SCs. We show the necessity of a divisibility constraint for the existence of an SC constructed by the popular diagonal embedding technique, that achieves this rate-bound under a stringent delay requirement. We then show that a construction known in the literature achieves this rate-bound when the divisibility constraint is met. We further show the equivalence of the SCs considered and $\text{SC}_{\text{ERR}}$s for the setting of multiple error bursts, under a stringent delay requirement.