🤖 AI Summary
This work investigates the information stability and existence of Shannon capacity for Markovian add-delete channels subject to synchronization errors (insertions/deletions) with memory. Specifically, it considers channels where errors are modeled by a stationary ergodic finite-state Markov chain. The paper extends information stability theory—previously established only for memoryless synchronous channels—to this class of Markovian insertion/deletion channels, constituting the first such generalization. By integrating ergodic theory, the asymptotic equipartition property (AEP), and stationarity of Markov chains, the authors construct a generalized typical set framework and develop joint typicality analysis to rigorously establish information stability. Consequently, the Shannon capacity is proven to exist for these channels. This result provides a foundational capacity-theoretic basis for practical memory-inclusive communication systems—such as DNA-based data storage—and furnishes rigorous theoretical guarantees for the design of capacity-approaching coding schemes.
📝 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.