Factor Graph Optimization of Error-Correcting Codes for Belief Propagation Decoding

TL;DR

This work proposes for the first time a gradient-based data-driven approach for the design of sparse codes and develops locally optimal codes with respect to Belief Propagation decoding via the learning of the Factor graph under channel noise simulations.

Abstract

The design of optimal linear block codes capable of being efficiently decoded is of major concern, especially for short block lengths. As near capacity-approaching codes, Low-Density Parity-Check (LDPC) codes possess several advantages over other families of codes, the most notable being its efficient decoding via Belief Propagation. While many LDPC code design methods exist, the development of efficient sparse codes that meet the constraints of modern short code lengths and accommodate new channel models remains a challenge. In this work, we propose for the first time a gradient-based data-driven approach for the design of sparse codes. We develop locally optimal codes with respect to Belief Propagation decoding via the learning of the Factor graph under channel noise simulations. This is performed via a novel complete graph tensor representation of the Belief Propagation algorithm, optimized over finite fields via backpropagation and coupled with an efficient line-search method. The proposed approach is shown to outperform the decoding performance of existing popular codes by orders of magnitude and demonstrates the power of data-driven approaches for code design.

Figures (12)

• Figure 1: For the Hamming(7,4) Code: (a) parity check matrix, the induced (b) Tanner graph, and (c) the corresponding unrolled Trellis graph with two iterations, with odd layers in blue and even layers in red. In (d) we present our approach for structure learning via the learned binary weighting of the edges of the complete bipartite factor graph unlike the conventional sparse representation.
• Figure 2: Statistics of improvement in dB for the (a) AWGN, (b) fading, and (c) bursting channel on the sparse codes only. We provide the mean and standard deviation as well as the minimum and maximum improvements.
• Figure 3: Performance of the method on random codes under different sparsity rate initialization $p$.
• Figure 4: Performance of the method on constrained systematic random codes under different sparsity rate initialization $p$ on the AWGN channel.
• Figure 5: Sparsity reduction of the proposed codes.