A Practical Indexing Scheme for Noisy Shuffling Channels Using Cosets of Polar Codes

The noisy shuffling channel models the conditions encountered in DNA storage systems, where transmitted data segments experience random permutation and substitution errors. Reliable communication over this channel requires effective indexing and channel coding strategies for segment order restoration and error correction. This paper introduces a concatenated coding approach for communication over the noisy shuffling channel using Reed-Solomon (RS) codes as outer codes and polar codes as inner codes. A coset-based indexing method, derived from polar codes, is proposed. A joint decoder is designed to detect the permutation pattern and perform polar decoding simultaneously. An upper bound on the frame error rate (FER) is derived when minimum distance decoding is employed for decoding. Also, an approximate analysis of the FER using random coding is conducted. A mapping between the cosets of the polar code and subsets of its frozen bits is established to design cosets achieving lower FERs compared to a commonly used explicit indexing method. Furthermore, a low-complexity decoding approach is devised, providing a trade-off between the computational complexity of the joint decoder and its performance.