Abstract
Permutation arrays under the Chebyshev metric have been considered for error correction in noisy channels. Let P(n, d) denote the maximum size of any array of permutations on n symbols with pairwise Chebyshev distance d. We give new techniques and improved upper and lower bounds on P(n, d), including a precise formula for P(n, 2).
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References
Bereg S., Bumpass W., Haghpanah M., Malouf B., Sudborough I.H.: Bounds for permutation arrays under Kendall tau metric. arXiv:2301.11423 (2023)
Bereg S., Levy A., Sudborough I.H.: Constructing permutation arrays from groups. Des. Codes Cryptogr. 86, 1095–1111 (2018). https://doi.org/10.1007/s10623-017-0381-1.
Bereg S., Miller Z., Mojica L.G., Morales L., Sudborough I.H.: New lower bounds for permutation arrays using contraction. Des. Codes Cryptogr. 87, 2105–2128 (2019).
Buzaglo S., Etzion T.: Bounds on the size of permutation codes with the Kendall \(\tau \)-metric. IEEE Trans. Inform. Theory 61, 3241–3250 (2015).
Chu W., Colbourn C.J., Dukes P.: Constructions for permutation codes in powerline communications. Des. Codes Cryptogr. 32, 51–64 (2004).
Deza M.M., Huang T.: Metrics on permutations, a survey. J. Comb. Inf. Syst. Sci. 23, 173–185 (1998).
Hagberg A.A., Schult D.A., Swart P.J.: Exploring network structure, dynamics, and function using networkx. In: Proceedings of the 7th Python in Science Conference SciPy2008, pp. 11–15 (2008).
Jiang A., Schwartz M., Bruck J.: Correcting charge-constrained errors in the rank-modulation scheme. IEEE Trans. Inform. Theory 56, 2112–2120 (2010).
Kløve T.: Lower bounds on the size of spheres of permutations under the Chebyshev distance. Des. Codes Cryptogr. 59, 183–191 (2011).
Kløve T., Lin T.-T., Tsai S.-C., Tzeng W.-G.: Permutation arrays under the Chebyshev distance. IEEE Trans. Inf. Theory 56, 2611–2617 (2010).
Rossi R.A., Gleich D.F., Gebremedhin A.H., Patwary M.M.: A fast parallel maximum clique algorithm for large sparse graphs and temporal strong components. arXiv:1302.6256 (2013)
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Communicated by T. Etzion.
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Bereg, S., Haghpanah, M., Malouf, B. et al. Improved bounds for permutation arrays under Chebyshev distance. Des. Codes Cryptogr. 92, 1023–1039 (2024). https://doi.org/10.1007/s10623-023-01326-1
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DOI: https://doi.org/10.1007/s10623-023-01326-1