Capacity of Additive-Noise Sticky Channels

2026-08-02

AI Summary

This study addresses the long-standing open problem of determining the capacity of the sticky channel with additive noise, which preserves run-length structure under synchronization errors yet has lacked a closed-form capacity expression. By reframing the problem as a memoryless channel capacity per unit cost over the non-negative integers, and leveraging information-theoretic analysis, the capacity-per-unit-cost framework, and run-length-constrained coding, the authors derive the exact capacity for Bernoulli noise parameter \( p \leq 1/2 \). They establish that the capacity remains constant over \( p \in [1/\varphi^2, 1/2] \), fully characterizing the capacity function in this interval, and provide tight analytical bounds for \( 1/2 < p < 1 \). The approach is further extended to general additive noise distributions, precisely identifying parameter regimes where zero-error run-length-constrained codes are suboptimal.

Abstract

Sticky channels, which never destroy nor create runs, are some of the simplest types of channels with synchronization errors (such as deletions, insertions, and replications). Despite their simplicity, we know little about the capacity of the sticky channels considered in the literature so far beyond what can be gleaned from purely numerical methods. Towards a broader and systematic exploration of these channels, we initiate the study of additive-noise sticky channels, a basic family of sticky channels that extend each input run by an amount independently sampled from a fixed noise distribution. The capacity of these channels corresponds to the capacity per unit cost of additive-noise memoryless channels over the non-negative integers, and they capture some aspects of the loss of synchronization caused by homopolymer length miscalls in DNA sequencing. We first focus on the setting of Bernoulli additive noise with parameter $p$, and uncover curious behavior of the capacity beyond what numerical methods can tell us. For example, the capacity is constant when $p\in [1/\varphi^2,1/2]$ with $\varphi\approx 1.618$ the golden ratio, achieved by zero-error runlength-constrained coding, and behaves differently when $p\approx 0$ vs. when $p\approx 1$. More generally, we determine the capacity exactly for all $p\leq 1/2$, and when $1/2<p<1$ we give analytical bounds that allow us to characterize the asymptotic behavior of the capacity as $p\to 1$. We also extend our analysis to the setting where input strings have bounded runlengths, motivated by DNA-based data storage. Then, we study additive-noise sticky channels beyond Bernoulli noise. We derive capacity lower bounds for all additive-noise sticky channels whose noise distributions have a given mean and support. These lower bounds allow us to characterize the regime where zero-error runlength-constrained coding is never optimal.

Problem

• sticky channels
• channel capacity
• synchronization errors
• additive noise
• runlength-constrained coding

Innovation

• additive-noise sticky channels
• channel capacity
• runlength-constrained coding
• synchronization errors
• DNA data storage