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., $n = 24$).

翻译：本文证明，线性码结合拒绝采样可构建一种模拟加性可交换噪声信道的容量可达方案。具体而言，该方案的通信量超出功能信息下界不超过 $\log e + 1$ 比特，因此可用于失真信源编码以实现率失真函数。我们讨论了基于BCH码与极化码的实用实现方案。对于二进制对称信道的仿真，采用分组长度 $n = 63$ 的BCH码构造方案，其速率可与 $n = 4096$ 的PolarSim方案相媲美，同时显著降低了延迟。基于极化码的构造方案以多项式平均复杂度渐近达到信道容量。此外，利用贪婪拒绝采样的思想，我们提出了一种基于任意线性码构建容量可达方案的算法。实验表明，在特定失真度范围内，我们的构造在汉明失真下的失真信源编码性能优于传统覆盖码，且即使在较小分组长度（如 $n = 24$）下仍表现良好。