Rate-Optimal Streaming Codes Under an Extended Delay Profile for Three-Node Relay Networks With Burst Erasures

TL;DR

This work addresses rate-optimal streaming over a three-node relay network under burst erasures with end-to-end delay $T$. It introduces an extended delay profile to achieve optimal rates under a relaxed condition $\frac{T-u-v}{2u-v} \le \left\lfloor \frac{T-u-v}{u} \right\rfloor$, generalizing prior divisibility constraints. The paper develops SR and RD code constructions under this framework and shows how to combine them to reach the non-adaptive rate bound for TBSCs, with explicit delay profiles and invertibility lemmas ensuring decodability by the deadline. It also discusses a broader, earlier construction (SR-2/RD) under a different regime and provides concrete examples (e.g., $b_1=3,b_2=4$ and $b_1=4,b_2=3$) to illustrate achievability and limitations, including cases where the optimal rate cannot be attained without adaptivity.

Abstract

This paper investigates streaming codes for three-node relay networks under burst packet erasures with a delay constraint $T$. In any sliding window of $T+1$ consecutive packets, the source-to-relay and relay-to-destination channels may introduce burst erasures of lengths at most $b_1$ and $b_2$, respectively. Let $u = \max\{b_1, b_2\}$ and $v = \min\{b_1, b_2\}$. Singhvi et al. proposed a construction achieving the optimal rate when $u\mid (T-u-v)$. In this paper, we present an extended delay profile method that attains the optimal rate under a relaxed constraint $\frac{T - u - v}{2u - v} \leq \left\lfloor \frac{T - u - v}{u} \right\rfloor$ and it strictly cover restriction $u\mid (T-u-v)$. %Furthermore, we demonstrate that the optimal rate for streaming codes is not achievable when $0< T-u-v<v$ under the convolutional code framework.

Figures (7)

• Figure 1: Comparison of valid parameter pairs for different constructions. Here, $b_1, b_2 \in [1, 5]$ and $b_1 + b_2 \leq T$. The total number of pairs for each $T$ is shown as a dashed black line. It can be observed that construction in this paper achieves full coverage in some cases, such as at $T=60$.
• Figure 2: Three-node relay networks with burst erasure $(b_1,b_2,T)$
• Figure 3: Structure of the encoding matrix $\mathbf{P}_{\hbox{\tiny SR}}$ for the SR code. The red blocks indicate assigned (non-zero) submatrices, while all other entries are zero. The matrix is clearly partitioned into $p+1$ block matrices $\mathbf{P}_0, \mathbf{P}_1, \ldots, \mathbf{P}_p$, which directly corresponds to the $p+1$ segments in the delay profile of the SR code.
• Figure 4: Structure of the encoding matrix $\mathbf{P}_{\hbox{\tiny SR-2}}$ for the SR-2 code. The red blocks indicate the positions containing assigned non-zero submatrices $P_i$, while all other entries are zero matrices.
• Figure 5: The messages inside the red dashed box are treated as linear combinations for transmission. Suppose a burst erasure of length 3 occurs, resulting in the loss of messages at times $0$, $1$, and $2$. Since messages before time $0$ and from time $3$ to $7$ are not lost, we can apply the decoding analysis from Example 1. Accordingly, the red marked symbols of $S[0]$ can be received with the delay profile $(7,7,7,7,6,3,3,3,3)$. Following the decoding analysis in Example 1, the combined symbol $(s_1[0], s_2[0], s_3[0], s_4[0], s_5[0], s_6[0], s_7[0] + s_5[-1], s_8[0] + s_5[-2], s_9[0])$ can be recovered by the known symbols at relay node, as specified by the same delay profile.

Theorems & Definitions (21)

• Remark 1
• Definition 1
• Lemma 1
• Definition 2
• Lemma 2
• Lemma 3
• Definition 3
• Lemma 4
• Theorem 5
• Lemma 6