During the project's first two years, we have made progress on several fronts, ranging from determining ultimate information-theoretic limits to practical coding solutions. Specifically,
1) We developed capacity bounds for the Poisson-Repeat channel, which is a model for synchronization errors developed previously in the literature. Our related paper (presented in IEEE ISIT 2023) showed that one can provide side information on the number of deletions in a given block and convert the Poisson-repeat channel to independent decoupled channels. Hence, it becomes possible to use numerical approaches to obtain the Poisson-repeat channel with side information, hence an upper bound on its capacity.
2) We developed a basic concatenated coding principle for DNA storage channels. Noisy shuffling channels are used as models for DNA storage channels. In such systems, the order in which the sequences are stored is lost - due to the nature of the DNA pool; hence, it becomes highly challenging to decode the stored information. Explicit indexing is a standard technique to alleviate the problems. In a paper we have presented in IEEE GLOBECOM 2023 (whose extended version has been accepted for publication in IEEE Trans. on Communications), we have proposed an effective alternative to explicit indexing. We showed that it is possible to use polar codes (or other block codes) with different cosets, with the specific cosets being used containing the order information. The proposed solution is highly practical and offers improved error probability performance compared to the other existing solutions.
3) We proved the information stability of insertion/deletion channels with memory, which is a model encountered in current DNA storage technologies. There is a classical result on the existence of Shannon capacity for memoryless synchronization error channels (under some mild conditions on the channel statistics). In our paper (presented in IEEE ISIT 2024, with the journal version being prepared), we proved that the information stability holds for the case with insertion/deletion processes possessing memory. Such errors with memory occur in practical DNA storage applications; hence, addressing the relevant channel models is critical. Our result establishes that Shannon capacity exists for these models as well. The methodology developed could be useful for generating information stability results for other communication schemes over non-traditional media. We also provide specific examples of deletion channels with Markov memory and numerically evaluate capacity bounds. The results allow us to quantify the capacity difference between memoryless deletion channels and those with memory with the same deletion probability.
4) We determined the asymptotic capacity of insertion channels (for different insertion models) for small insertion probabilities. We obtained the dominant terms of the channel capacity; hence, we can characterize the capacity of insertion channels (which are valuable models for various applications, including DNA storage) in the asymptotic regime. We expect the general methodology to be crucial in investigating other channel models with synchronization errors, including channels exhibiting both deletions and insertions and those exhibiting insertions/deletions/substitutions. The extended version of the paper has been completed and submitted for publication in IEEE Transactions on Information Theory.
5) We explored and developed deep-learning-based decoding algorithms for concatenated coding using marker codes as inner codes for insertion and deletion channels. We considered concatenated coding approaches (where the outer code can be a low-density parity-check code or a convolutional code, and the inner code is a marker code). Such coding solutions provide the state of the art of communicating over channels with synchronization errors. However, the decoding algorithms are highly complex, which hinders their practical use. Motivated by this, and with the premise that the proposed solutions could be promising for more complicated models, too, we developed deep-learning architectures for decoding of concatenated coding solutions over insertion/deletion channels.
