Two-Insertion/Deletion/Substitution Correcting Codes (2401.11231v1)
Abstract: In recent years, the emergence of DNA storage systems has led to a widespread focus on the research of codes correcting insertions, deletions, and classic substitutions. During the initial investigation, Levenshtein discovered the VT codes are precisely capable of correcting single insertion/deletion and then extended the VT construction to single-insertion/deletion/substitution ($1$-ins/del/sub) correcting codes. Inspired by this, we generalize the recent findings of $1$-del $1$-sub correcting codes with redundancy $6\log_{2}n+O(1)$ to more general $2$-ins/del/sub correcting codes without increasing the redundancy. Our key technique is to apply higher-order VT syndromes to distinct objects and accomplish a systematic classification of all error patterns.
- K. A. S. Abdel-Ghaffar, F. Palunčić, H. C. Ferreira, and W. A. Clarke, “On Helberg’s generalization of the Levenshtein code for multiple deletion/insertion error correction,” IEEE Trans. Inf. Theory, vol. 58, no. 3, pp. 1804–1808, 2012.
- J. Brakensiek, V. Guruswami, and S. Zbarsky, “Efficient low-redundancy codes for correcting multiple deletions,” IEEE Trans. Inf. Theory, vol. 64, no. 5, pp. 3403–3410, 2018.
- R. Gabrys and F. Sala, “Codes correcting two deletions,” IEEE Trans. Inf. Theory, vol. 65, no. 2, pp. 965–974, 2019.
- R. Gabrys, V. Guruswami, J. Ribeiro, and K. Wu, “Beyond single-deletion correcting codes: substitutions and transpositions,” IEEE Trans. Inf. Theory, vol. 69, no. 1, pp. 169–186, 2023.
- V. Guruswami and J. Håstad, “Explicit two-deletion codes with redundancy matching the existential bound,” IEEE Trans. Inf. Theory, vol. 67, no. 10, pp. 6384–6394, 2021.
- R. Heckel, G. Mikutis, and R. N. Grass, “A characterization of the DNA data storage channel,” Sci. Rep., vol. 9, no. 1, p. 9663, 2019.
- A. S. J. Helberg and H. C. Ferreira, “On multiple insertion/deletion correcting codes,” IEEE Trans. Inf. Theory, vol. 48, no. 1, pp. 305–308, 2002.
- V. I. Levenshtein, “Binary codes capable of correcting deletions, insertions, and reversals,” Soviet Phys. Doklady, vol. 10, no. 8, pp. 707–710, 1966.
- V. I. Levenshtein, “Asymptotically optimum binary code with correction for losses of one or two adjacent bits,” Problems of Cybernetics, vol. 19, pp. 298–304, 1967.
- F. Palunčić, T. G. Swart, J. H. Weber, H. C. Ferreira, and W. A. Clarke, “A note on non-binary multiple insertion/deletion correcting codes,” in Proc. IEEE Inf. Theory Workshop (ITW), pp. 683–687, 2011.
- F. Palunčić, K. A. S. Abdel-Ghaffar, H. C. Ferreira, and W. A. Clarke, “A multiple insertion/deletion correcting code for run-length limited sequences,” IEEE Trans. Inf. Theory, vol. 58, no. 3, pp. 1809–1824, 2012.
- J. Sima, N. Raviv, and J. Bruck, “Two deletion correcting codes from indicator vectors,” IEEE Trans. Inf. Theory, vol. 66, no. 4, pp. 2375–2391, 2020.
- J. Sima and J. Bruck, “On optimal k-deletion correcting codes,” IEEE Trans. Inf. Theory, vol. 67, no. 6, pp. 3360–3375, 2021.
- I. Smagloy, L. Welter, A. Wachter-Zeh, and E. Yaakobi, “Single-deletion single-substitution correcting codes,” IEEE Trans. Inf. Theory, vol. 69, no. 12, pp. 7659–7671, 2023.
- W. Song, K. Cai, and T. T. Nguyen, “List-decodable codes for single-deletion single-substitution with list-size two,” in Proc. IEEE Int. Symp. Inf. Theory (ISIT), pp. 1004–1009, 2022.
- W. Song, N. Polyanskii, K. Cai, and X. He, “Systematic codes correcting multiple-deletion and multiple-substitution errors,” IEEE Trans. Inf. Theory, vol. 68, no. 10, pp. 6402–6416, 2022.
- R. R. Varshamov and G. M. Tenengolts, “Codes which correct single asymmetric errors,” Avtomatica i Telemekhanika, vol. 26, no. 2, pp. 288–292, 1965.