This paper introduces a neural polar decoder (NPD) for deletion channels with a constant deletion rate. Existing polar decoders for deletion channels exhibit high computational complexity of $O(N^4)$, where $N$ is the block length. This limits the application of polar codes for deletion channels to short-to-moderate block lengths. In this work, we demonstrate that employing NPDs for deletion channels can reduce the computational complexity. First, we extend the architecture of the NPD to support deletion channels. Specifically, the NPD architecture consists of four neural networks (NNs), each replicating fundamental successive cancellation (SC) decoder operations. To support deletion channels, we change the architecture of only one. The computational complexity of the NPD is $O(AN\log N)$, where the parameter $A$ represents a computational budget determined by the user and is independent of the channel. We evaluate the new extended NPD for deletion channels with deletion rates $δ\in\{0.01, 0.1\}$ and we verify the NPD with the ground truth given by the trellis decoder by Tal et al. We further show that due to the reduced complexity of the NPD, we are able to incorporate list decoding and further improve performance. We believe that the extended NPD presented here could have applications in future technologies like DNA storage.

Synchronization errors, such as insertions and deletions, present a fundamental challenge in DNA-based data storage systems, arising from both synthesis and sequencing noise. These channels are often modeled as insertion-deletion-substitution (IDS) channels, for which designing maximum-likelihood decoders is computationally expensive. In this work, we propose a data-driven approach based on neural polar decoders (NPDs) to design low-complexity decoders for channels with synchronization errors. The proposed architecture enables decoding over IDS channels with reduced complexity $O(AN log N )$, where $A$ is a tunable parameter independent of the channel. NPDs require only sample access to the channel and can be trained without an explicit channel model. Additionally, NPDs provide mutual information (MI) estimates that can be used to optimize input distributions and code design. We demonstrate the effectiveness of NPDs on both synthetic deletion and IDS channels. For deletion channels, we show that NPDs achieve near-optimal decoding performance and accurate MI estimation, with significantly lower complexity than trellis-based decoders. We also provide numerical estimates of the channel capacity for the deletion channel. We extend our evaluation to realistic DNA storage settings, including channels with multiple noisy reads and real-world Nanopore sequencing data. Our results show that NPDs match or surpass the performance of existing methods while using significantly fewer parameters than the state-of-the-art. These findings highlight the promise of NPDs for robust and efficient decoding in DNA data storage systems.

We construct pseudorandom error-correcting codes (or simply pseudorandom codes), which are error-correcting codes with the property that any polynomial number of codewords are pseudorandom to any computationally-bounded adversary. Efficient decoding of corrupted codewords is possible with the help of a decoding key. We build pseudorandom codes that are robust to substitution and deletion errors, where pseudorandomness rests on standard cryptographic assumptions. Specifically, pseudorandomness is based on either $2^{O(\sqrt{n})}$-hardness of LPN, or polynomial hardness of LPN and the planted XOR problem at low density. As our primary application of pseudorandom codes, we present an undetectable watermarking scheme for outputs of language models that is robust to cropping and a constant rate of random substitutions and deletions. The watermark is undetectable in the sense that any number of samples of watermarked text are computationally indistinguishable from text output by the original model. This is the first undetectable watermarking scheme that can tolerate a constant rate of errors. Our second application is to steganography, where a secret message is hidden in innocent-looking content. We present a constant-rate stateless steganography scheme with robustness to a constant rate of substitutions. Ours is the first stateless steganography scheme with provable steganographic security and any robustness to errors.