Efficient decoding of permutation codes in the Hamming metric

Develop an efficient decoding algorithm for the Hamming-metric permutation codes required by the stable-deletion-correcting permutation-code construction used in the constructed multiplicity-free deletion-correcting code.

Background

The stable-deletion-correcting permutation-code construction cited by the paper relies on a decoding algorithm for permutation codes in the Hamming metric. The paper notes that the best known Hamming-metric permutation codes do not currently have efficient decoding algorithms, which prevents the construction from directly providing the desired efficient decoding implementation. The authors explicitly list efficient Hamming-metric decoding of permutation codes as an important open problem.

References

Important open problems include finding efficient encoding and message recovery algorithms for our code and the decoding of permutation codes in the Hamming metric.

— A New Construction of Non-Binary Deletion Correcting Codes and their Decoding  (2501.13534 - Schaller et al., 23 Jan 2025) in Section Conclusion and Discussion; related discussion in Section 3.1