Rejection-Sampled Linear Codes for Lossy Compression and Channel Simulation

TL;DR

This paper introduces a novel scheme combining linear codes and rejection sampling to achieve capacity for simulating additive exchangeable noise channels, with applications in lossy source coding and practical implementations using BCH and polar codes.

Contribution

It presents a capacity-achieving scheme for channel simulation and lossy compression using linear codes with rejection sampling, including practical algorithms and performance analysis.

Findings

Achieves near-optimal communication rates within a small gap of the lower bound.

BCH-based construction with small blocklength outperforms larger polar code schemes in latency.

Polar-based construction asymptotically reaches channel capacity with polynomial complexity.

Abstract

We show that linear codes combined with rejection sampling can yield a capacity-achieving scheme for simulating additive exchangeable noise channels. Specifically, our scheme achieves an amount of communication within $\log e + 1$ bits from the excess functional information lower bound. Hence, it can be used in lossy source coding to achieve the rate-distortion function. We discuss practical implementations based on BCH codes and polar codes. For the simulation of binary symmetric channels, the BCH-based construction with a blocklength of $n = 63$ attains a rate comparable to the PolarSim with $n = 4096$, while significantly reducing the latency. The polar-based construction asymptotically achieves the channel capacity with polynomial average complexity. Furthermore, using the idea from greedy rejection sampling, we propose an algorithm to construct capacity-achieving schemes based on any linear codes. Experiments reveal that our construction can outperform conventional covering codes for lossy source coding with Hamming distortion for a certain range of distortion levels, and performs well even when the blocklength is small (e.g., $n = 24$).