Streaming codes are a class of packet-level erasure codes that are designed with the goal of ensuring recovery in low-latency fashion, of erased packets over a communication network. It is well-known in the streaming code literature, that diagonally embedding codewords of a $[\tau+1,\tau+1-a]$ Maximum Distance Separable (MDS) code within the packet stream, leads to rate-optimal streaming codes capable of recovering from $a$ arbitrary packet erasures, under a strict decoding delay constraint $\tau$. Thus MDS codes are geared towards the efficient handling of the worst-case scenario corresponding to the occurrence of $a$ erasures. In the present paper, we have an increased focus on the efficient handling of the most-frequent erasure patterns. We study streaming codes which in addition to recovering from $a>1$ arbitrary packet erasures under a decoding delay $\tau$, have the ability to handle the more common occurrence of a single-packet erasure, while incurring smaller delay $r<\tau$. We term these codes as $(a,\tau,r)$ locally recoverable streaming codes (LRSCs), since our single-erasure recovery requirement is similar to the requirement of locality in a coded distributed storage system. We characterize the maximum possible rate of an LRSC by presenting rate-optimal constructions for all possible parameters $\{a,\tau,r\}$. Although the rate-optimal LRSC construction provided in this paper requires large field size, the construction is explicit. It is also shown that our $(a,\tau=a(r+1)-1,r)$ LRSC construction provides the additional guarantee of recovery from the erasure of $h, 1 \leq h \leq a$, packets, with delay $h(r+1)-1$. The construction thus offers graceful degradation in decoding delay with increasing number of erasures.
翻译:Streaming 代码是用于确保在一个通信网络上以低延迟速度恢复被擦掉的包包的包级删除代码( 串流代码), 目的是确保以低延迟速度恢复在通信网络上被擦掉的包级删除代码。 在流化代码文献中, 将 $\ tau+1,\ tau+1- a] 的最大距离分解( MDS) 代码直接嵌入 $\\ tau, 导致以 $ 任意的包级删除代码恢复。 在严格解码延迟参数下, $\ t$ 。 因此, MDRS 代码用于高效处理最坏的情景, 与 $ + 折现的代码相对应。 我们的研究流化代码除了从 $> 1 任意的包级删除代码中恢复 $\ tcol- lax lax lax lax r= droad mail mail lax lax lax the more liversal lax listal listrue listal lax lax as lax a lax lax lax the lax lax lax lax lax lax lax lax lax lax lax lax lax lax lax laut lautt et) 后, 我们 后, 我们 laxxxxx 以 以 以 美元 以 美元 美元 美元 美元