On Streaming Codes for Simultaneously Correcting Burst and Random Erasures

TL;DR

This paper develops optimal streaming codes capable of correcting both burst and random erasures within strict delay constraints, providing new constructions and bounds on field size and code parameters.

Contribution

It introduces a diagonal embedding-based construction for streaming codes over sliding-window channels with optimal rate and analyzes field size limitations under the MDS conjecture.

Findings

Optimal rate of streaming codes determined for the SW channel model.

Construction of codes with sub-linear field size for certain parameters.

Characterization of cyclic codes achieving bounds on minimum distance.

Abstract

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.