AI 中文总结
本文研究加性噪声粘性信道的容量，确定$p\leq1/2$时的精确容量、$1/2<p<1$时的解析界，推导伯努利外噪声的容量下界，还扩展到有界游程长度场景，为DNA数据存储等应用提供理论支撑。
AI 中文摘要
粘性信道是一类不会破坏也不会生成游程的信道，属于带同步错误（如删除、插入和复制）的最简单信道类型。尽管结构简单，但目前文献中对粘性信道容量的认知仅局限于纯数值方法能获取的内容。为更广泛、系统地探索这类信道，本文启动对加性噪声粘性信道的研究，该类信道是粘性信道的基本家族，会将每个输入游程按固定噪声分布独立采样得到的量进行扩展。这类信道的容量对应非负整数上加性噪声无记忆信道的单位成本容量，且能捕捉DNA测序中同聚物长度误判导致的部分同步损失。我们首先聚焦参数为$p$的伯努利加性噪声场景，揭示了数值方法无法得出的容量特殊行为：例如当$p\in [1/\varphi^2,1/2]$（其中$\varphi\approx1.618$为黄金分割比）时，容量为常数，由零错误游程长度受限编码实现；当$p\approx0$与$p\approx1$时，容量表现不同。更一般地，我们确定了所有$p\leq1/2$时的精确容量，对$1/2<p<1$的情况给出解析界，可刻画$p\to1$时容量的渐近行为；受DNA数据存储的启发，我们还将分析扩展到输入字符串具有有界游程长度的场景。随后，我们研究伯努利噪声之外的加性噪声粘性信道，对所有噪声分布具有给定均值和支撑的加性噪声粘性信道，推导了容量下界，这些下界可刻画零错误游程长度受限编码永远非最优的区域。
英文摘要
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 $p\in [1/φ^2,1/2]$ with $φ\approx 1.618$ 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.
CommentsThis is the extended version of a paper accepted to appear at the 2026 IEEE Information Theory Workshop (ITW 2026)
