On Streaming Codes for Simultaneously Correcting Burst and Random Erasures
<|reference_start|>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 $\le b$ 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.<|reference_end|>
@article{bhatnagar2024on,
    title={On Streaming Codes for Simultaneously Correcting Burst and Random
  Erasures},
    author={Shobhit Bhatnagar, Biswadip Chakraborty and P. Vijay Kumar},
    journal={arXiv preprint arXiv:2405.06621},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.06621},
    primaryClass={cs.IT math.IT}
}