Protograph-Based Batched Network Codes

TL;DR

The work proposes protograph-based batched network codes (P-BNCs) to substantially improve finite-length BP decoding performance in networks with erasures. By integrating a sparse LDPC-like precode with a BATS-code-like protograph and employing a two-step lifting plus puncturing, P-BNCs achieve rate-compatibility and robust decoding across varying channel conditions. The authors develop a protograph-based density evolution framework and define a decoding-threshold objective to guide optimization, demonstrating that optimized P-BNCs can closely approach the finite-length ML limits while outperforming conventional BATS codes in representative line-network scenarios. This approach offers a practical, scalable path to high-throughput reliable multicast in networks with diverse erasure profiles.

Abstract

Batched network codes (BNCs) are a low-complexity solution for communication through networks with packet loss. Although their belief propagation (BP) performance is proved to approach capacity in the asymptotic regime, there is no evidence indicating that their BP performance is equally good in the finite-length regime. In this paper, we propose a protograph-based construction for BNCs, referred to as protograph-based BNCs (P-BNCs), which significantly differs from existing BNCs in three aspects: 1) The vast majority of existing construction methods mainly focus on the degree distribution of check nodes (CNs), whereas P-BNCs not only specify the degree distributions of CNs and variable nodes (VNs) but also partially constrain the connectivity between CNs and VNs. 2) Traditional BNCs use a fixed degree distribution to generate all batches, making their performance highly sensitive to channel conditions, but P-BNCs achieve good performance under varying channel conditions due to their rate-compatible structures. 3) The construction of PBNCs takes into account joint BP decoding with a sparse precode, whereas traditional constructions typically do not consider a precode, or assume the presence of a precode that can recover a certain fraction of erasures. Thanks to these three improvements, P-BNCs not only have higher achievable rates under varying channel conditions, but more importantly, their BP performance is significantly improved at practical lengths.

Figures (10)

• Figure 1: Achievable rates of the standard BATS code and the P-BNC over a line network with two channels. The two erasure probabilities of two channels are supposed to be the same. The batch size is $16$ (i.e., each batch has 16 coded packets). The standard BATS code is optimized for the erasure probability varying from $0.1$ to $0.6$ using BATS. The rate is defined as $A/N$, where $A$ is the number of input packets and $N$ is the number of batches. To evaluate the performance at a practical length, we set $A = 1600$, and $N$ is chosen based on simulation at which the frame error rate reaches $0.1$.
• Figure 2: An example of lifting a protograph with $Z = 2$. The figure is referenced from 7112076.
• Figure 3: The Tanner graph of a BNC with a $(2,5)$-regular LDPC precode.
• Figure 4: The protograph of $\bf B$ in Example \ref{['example:protograph']}.
• Figure 5: A tree ${\cal T}_{1}(2)$ from the protograph in Example \ref{['example:protograph']}. The root is a type-2 VN and the height is 1. The messages used in density evolution are labeled on the edges.

Theorems & Definitions (19)

• Example 1
• Lemma 1
• Example 2: LDPC-Chunked Codes Tang2018Lchunked
• Example 3: Overlapped Chunked Codes overlapChunk and Gamma Codes GammaCodes2012
• Example 4
• Definition 1
• Remark 1
• Remark 2
• Definition 2: Tree from node perspective
• Theorem 1