Smoothing of binary codes, uniform distributions, and applications

TL;DR

This work develops a comprehensive smoothing framework for binary codes under additive noise, tying output distributions to uniformity via Rényi divergences $D_{\alpha}$ and linking smoothing to channel resolvability. It derives asymptotic smoothing thresholds: the $D_{\alpha}$-smoothing capacity $S_{\alpha}^{r}$ equals $1-\pi(\alpha)$ for $\alpha>1$, with Bernoulli and ball noise yielding explicit capacity formulas and rate regions. The authors identify explicit code families, notably RM codes and polar constructions, that achieve smoothing capacities or secrecy guarantees in wiretap settings, and they establish deep connections between smoothing, error correction, and secrecy via duality and erasure-coding results. The finite-case discussion shows uniformly packed codes can be perfectly smoothed by appropriately chosen radial kernels, linking geometric packing properties to information-theoretic smoothing and practical decoding implications. Together, these results illuminate when smoothing to uniformity is possible, quantify the required code rates, and provide explicit constructions with practical implications for secure communication and reliable decoding.

Abstract

The action of a noise operator on a code transforms it into a distribution on the respective space. Some common examples from information theory include Bernoulli noise acting on a code in the Hamming space and Gaussian noise acting on a lattice in the Euclidean space. We aim to characterize the cases when the output distribution is close to the uniform distribution on the space, as measured by Rényi divergence of order $α\in (1,\infty]$. A version of this question is known as the channel resolvability problem in information theory, and it has implications for security guarantees in wiretap channels, error correction, discrepancy, worst-to-average case complexity reductions, and many other problems. Our work quantifies the requirements for asymptotic uniformity (perfect smoothing) and identifies explicit code families that achieve it under the action of the Bernoulli and ball noise operators on the code. We derive expressions for the minimum rate of codes required to attain asymptotically perfect smoothing. In proving our results, we leverage recent results from harmonic analysis of functions on the Hamming space. Another result pertains to the use of code families in Wyner's transmission scheme on the binary wiretap channel. We identify explicit families that guarantee strong secrecy when applied in this scheme, showing that nested Reed-Muller codes can transmit messages reliably and securely over a binary symmetric wiretap channel with a positive rate. Finally, we establish a connection between smoothing and error correction in the binary symmetric channel.

Figures (2)

• Figure 1: Capacities and achievable rates for perfect smoothing. The lowermost curve gives Shannon capacity of the BSC$(\delta)$, the second curve from the bottom is the smoothing threshold for duals of BEC capacity-achieving codes, the third one is $S_2^{\beta_\delta}$ and the top one is $S_\infty^{\beta_\delta}.$
• Figure 2: Achievable rates in the BSC wiretap channel with BEC capacity-achieving codes. The bottom curve is the lower bound on the code rate that guarantees decodability on a BSC$(\delta)$. The middle curve shows Shannon's capacity and the top one is the $D_1$-smoothing threshold for Bernoulli noise $T_{\delta}.$

Theorems & Definitions (74)

• Remark 1
• Theorem 2.1: han1993approximationhayashi2006general
• Theorem 2.2: yu2018renyi, Theorem 2
• Corollary 2.3: yu2018renyi, Eq. (55)
• Definition 3.1
• Proposition 3.1
• proof
• Definition 3.2
• Definition 3.3
• Lemma 3.2