With the emergence of new storage and communication methods, the insertion, deletion, and substitution (IDS) channel has attracted considerable attention. However, many topics on the IDS channel and the associated Levenshtein distance remain open, making the invention of a novel IDS-correcting code a hard task. Furthermore, current studies on single-IDS-correcting code misalign with the requirements of applications which necessitates the correcting of multiple errors. Compromise solutions have involved shortening codewords to reduce the chance of multiple errors. However, the code rates of existing codes are poor at short lengths, diminishing the overall storage density. In this study, a novel method is introduced for designing high-code-rate single-IDS-correcting codewords through deep Levenshtein distance embedding. A deep learning model is utilized to project the sequences into embedding vectors that preserve the Levenshtein distances between the original sequences. This embedding space serves as a proxy for the complex Levenshtein domain, within which algorithms for codeword search and segment correcting is developed. While the concept underpinning this approach is straightforward, it bypasses the mathematical challenges typically encountered in code design. The proposed method results in a code rate that outperforms existing combinatorial solutions, particularly for designing short-length codewords.

Efficient computation or approximation of Levenshtein distance, a widely-used metric for evaluating sequence similarity, has attracted significant attention with the emergence of DNA storage and other biological applications. Sequence embedding, which maps Levenshtein distance to a conventional distance between embedding vectors, has emerged as a promising solution. In this paper, a novel neural network-based sequence embedding technique using Poisson regression is proposed. We first provide a theoretical analysis of the impact of embedding dimension on model performance and present a criterion for selecting an appropriate embedding dimension. Under this embedding dimension, the Poisson regression is introduced by assuming the Levenshtein distance between sequences of fixed length following a Poisson distribution, which naturally aligns with the definition of Levenshtein distance. Moreover, from the perspective of the distribution of embedding distances, Poisson regression approximates the negative log likelihood of the chi-squared distribution and offers advancements in removing the skewness. Through comprehensive experiments on real DNA storage data, we demonstrate the superior performance of the proposed method compared to state-of-the-art approaches.