Performance Bounds and Degree-Distribution Optimization of Finite-Length BATS Codes

TL;DR

The paper addresses finite-length performance of precoded BATS codes under joint BP and ML decoding by deriving two computable upper bounds on word-error probability, enabling principled degree-distribution optimization. It extends BATS codes with LDPC (or generic) precoding and analyzes decoding on a joint Tanner graph, providing bounds that illuminate decoding failure structures and codifies a practical optimization framework. The results show that optimized LDPC-BATS codes can substantially reduce transmission overhead (e.g., to under 50% of standard BATS in line networks) while keeping decoding complexity manageable via joint BP/ML or inactivation decoding. This work offers a first analytical treatment of BP and ML decoding for precoded BATS codes in the finite-length regime and delivers a concrete method to design more efficient codes for networks with packet loss.

Abstract

Batched sparse (BATS) codes were proposed as a reliable communication solution for networks with packet loss. In the finite-length regime, the error probability of BATS codes under belief propagation (BP) decoding has been studied in the literature and can be analyzed by recursive formulae. However, all existing analyses have not considered precoding or have treated the BATS code and the precode as two separate entities. In this paper, we analyze the word-wise error probability of finite-length BATS codes with a precode under joint decoding, including BP decoding and maximum-likelihood (ML) decoding. The joint BP decoder performs peeling decoding on a joint Tanner graph constructed from both the BATS and the precode Tanner graphs, and the joint ML decoder solves a single linear system with all linear constraints implied by the BATS code and the precode. We derive closed-form upper bounds on the error probability for both decoders. Specifically, low-density parity-check (LDPC) precodes are used for BP decoding, and any generic precode can be used for ML decoding. Even for BATS codes without a precode, the derived upper bound for BP decoding is more accurate than the approximate recursive formula, and easier to compute than the exact recursive formula. The accuracy of the two upper bounds has been verified by many simulation results. Based on the two upper bounds, we formulate an optimization problem to optimize the degree distribution of LDPC-precoded BATS codes, which improves BP performance, ML performance, or both. In our experiments, to transmit 128 packets over a line network with packet loss, the optimized LDPC-precoded BATS codes reduce the transmission overhead to less than 50% of that of standard BATS codes under comparable decoding complexity constraints.

Figures (14)

• Figure 1: Tanner graphs of a standard BATS code. (a) Encoding perspective. (b) Decoding perspective.
• Figure 2: The Tanner graph of an LDPC-BATS code with a $(2,5)$-regular LDPC precode.
• Figure 3: The error probabilities of the standard BATS code ${\mathscr{C}}_{\rm std}^{(256)}(128,n,16,{\bf \Psi}_1,{\bf h}_1)$ under BP decoding. The error probabilities are computed by the two recursive formulae in FL_analysis_BATS and Theorem \ref{['theorem_LDPC_BATS_BP']}.
• Figure 4: The error probabilities of the standard BATS code ${\mathscr{C}}_{\rm std}^{(256)}(256,n,16,{\bf \Psi}_2,{\bf h}_1)$ under BP decoding. The error probabilities are computed by the two recursive formulae in FL_analysis_BATS and Theorem \ref{['theorem_LDPC_BATS_BP']}.
• Figure 5: The simulation results and the upper bounds (Theorem \ref{['theorem_LDPC_BATS_BP']}) for the LDPC-BATS codes ${\mathscr{C}}_{\rm pre}^{(256)}(128,n,16,{\bf \Psi}_3/{\bf \Psi}_{17},{\bf h}_1,3,6,1/4)$ and $\tilde{\mathscr{C}}_{\rm pre}^{(256)}(128,n,16,{\bm \Psi}_3,{\bf h}_1,3,6)$ under BP decoding.

Theorems & Definitions (55)

• Definition 1: BATS Ensemble
• Definition 2: LDPC-BATS Ensemble
• Lemma 1: Weight_distribution_LDPC
• Theorem 1
• proof
• Remark 1
• Theorem 2
• proof
• Remark 2
• Remark 3