Rejection-Sampled Linear Codes for Lossy Compression and Channel Simulation

TL;DR

This work advances channel simulation and lossy source coding by pairing linear inner codes with rejection sampling to realize exact AXN-channel simulation with near-capacity efficiency. It introduces stochastic, state-dependent, and greedy RSSE schemes, providing both short-blocklength BCH-based implementations and asymptotic polar-based schemes, each with provable bounds and practical complexity. By exploiting RTC-decomposition and code-distance properties, the approach achieves performance close to the excess functional information lower bound and demonstrates competitive rate-distortion behavior at small blocklengths, often outperforming conventional covering codes. The framework also supports oblivious relaying and offers a constructive, scalable path to capacity-achieving schemes using structured codes.

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., $n = 24$).

Figures (6)

• Figure 1: Top: An oblivious relay applying channel simulation to simulate a noisy channel $p_{U|Y}$. Bottom: The effect of channel simulation is that the simulated output $\mathbf{U}$ is distributed as if it is the output of $\mathbf{Y}$ passed through another noisy channel $p_{U|Y}$.
• Figure 2: One-shot channel simulation.
• Figure 3: The communication rate and expected number of iterations given by the $S^*$-dependent BCH-RSSE for simulating $n$ copies of $\mathrm{BSC}(\alpha)$ with $n=31, 63$ and $\alpha\in[0,\frac{1}{2}]$.
• Figure 4: The communication rate needed by various algorithms to simulate the memoryless binary symmetric channel with $n=24$ and $\alpha\in[0,1/2]$.
• Figure 5: The communication rate needed by various algorithms to simulate the Hamming ball channel with $n=24$ and relative weights $\alpha=w/n$ for $w=0,\ldots,12$.

Theorems & Definitions (18)

• Definition 1
• Proposition 2
• Definition 3
• Proposition 4
• Lemma 5
• Corollary 6
• Example 1
• Remark 7
• Definition 8
• Definition 9