Markov Insertion/Deletion Channels: Information Stability and Capacity Bounds

TL;DR

The paper proves that insertion/deletion channels with memory governed by a stationary ergodic Markov chain are information-stable, ensuring the existence of Shannon capacity for the resulting channel sequence and ${\mathbf C}(V)={\mathbf I}(V)$. It introduces a modular, function-based approach to transform channels without changing capacity properties, enabling a robust proof strategy via a quasi-subadditive argument. The main results establish the existence of the i-capacity for Markov-IDS channels, its independence from the initial state, and, under bounded output lengths, information stability that yields equal coding and mutual information capacity. Numerical upper bounds for a two-state Markov deletion model demonstrate that memory in the deletion process can increase capacity bounds relative to i.i.d. deletions, with implications for DNA storage and synchronization-error systems.

Abstract

We consider channels with synchronization errors modeled as insertions and deletions. A classical result for such channels is their information stability, hence the existence of the Shannon capacity, when the synchronization errors are memoryless. In this paper, we extend this result to the case where the insertions and deletions have memory. Specifically, we assume that the synchronization errors are governed by a stationary and ergodic finite state Markov chain, and prove that such channel is information-stable, which implies the existence of a coding scheme which achieves the limit of mutual information. This result implies the existence of the Shannon capacity for a wide range of channels with synchronization errors, with different applications including DNA storage. The methods developed may also be useful to prove other coding theorems for non-trivial channel sequences.

Figures (4)

• Figure 1: Exemplary synchronization error channel with memory: deletion channel with two states.
• Figure 2: The upper bound on capacity for $\delta=0.01$, BDC capacity $\leq 0.945$.
• Figure 3: The upper bound on capacity for $\delta=0.1$, BDC capacity $\leq 0.689$.
• Figure 4: The upper bound on capacity for $\delta=0.5$, BDC capacity $\leq 0.213$.

Theorems & Definitions (18)

• Proposition 1: Theorems 3.1 and 3.2 in hu62shannon
• Proposition 2: A sufficient condition for strict information stability
• proof
• Proposition 3: Lemma 9 in mao18models
• proof
• Theorem 1: Existence of i-capacity
• Theorem 2: Information stability of the Markov-IDS channel sequence
• Proposition 4: Equation (4.3) in dobrushin67shannons
• Proposition 5: Function with small preimage
• proof