Streaming Codes for Handling a Combination of Burst and Random Erasures

Streaming codes may be regarded as packet-level convolutional codes that guarantee recovery from packet erasures under a strict decoding-delay constraint and are hence relevant to the low-latency objective of many modern communication systems. Past study of these codes has focused on performance over a tractable approximation of the Gilbert-Elliott channel model, known as the delay-constrained sliding window (DCSW) channel model. Under the DCSW channel model, within any sliding window of length w there can either be (i) a burst of at most b packet erasures or (ii) at most a random packet erasures. We study here, an extended version of the first constraint which permits e random erasures in addition to a burst of b erasures. We show that the capacity of this extended DCSW channel is strictly less than that of the corresponding DCSW channel in which b is replaced by $b+e$. Cyclic codes are easy to implement and are inherently well-suited to burst-erasure recovery. We identify a necessary and sufficient condition on the parity polynomial of an $[n, k]$ cyclic code that allows the code to recover from any burst of $n-k-s$ erasures along with any $\rho$ random erasures, $1 \leq \rho \leq s \leq n-k$. We use this result to construct cyclic codes that provide reliable communication over the extended DCSW channel for certain parameters.