Abstract
Permutation codes under different metrics have been extensively studied due to their potentials in various applications. Generalized Cayley metric is introduced to correct generalized transposition errors, including previously studied metrics such as Kendall’s \(\tau \)-metric, Ulam metric and Cayley metric as special cases. Since the generalized Cayley distance between two permutations is not easily computable, Yang et al. introduced a related metric of the same order, named the block permutation metric. Given positive integers n and d, let \(\mathcal {C}_{B}(n,d)\) denote the maximum size of a permutation code in \(S_n\) with minimum block permutation distance d. In this paper, we focus on the theoretical bounds of \(\mathcal {C}_{B}(n,d)\) and the constructions of permutation codes under block permutation metric. Using a graph theoretic approach, we improve the Gilbert–Varshamov type bound by a factor of \(\Omega (\log {n})\), when d is fixed and n goes into infinity. We also propose a new encoding scheme based on binary constant weight codes. Moreover, an upper bound beating the sphere-packing type bound is given when d is relatively close to n.
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The authors express their gratitude to the anonymous reviewer for the detailed and constructive comments which are very helpful to the improvement of the presentation of this paper.
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Communicated by J. H. Koolen.
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Y. Zhang: Supported in part by a Technion fellowship. G. Ge: Research supported by the National Natural Science Foundation of China under Grant Nos. 11431003 and 61571310, Beijing Scholars Program, Beijing Hundreds of Leading Talents Training Project of Science and Technology, and Beijing Municipal Natural Science Foundation.
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Xu, Z., Zhang, Y. & Ge, G. New theoretical bounds and constructions of permutation codes under block permutation metric. Des. Codes Cryptogr. 87, 2625–2637 (2019). https://doi.org/10.1007/s10623-019-00641-w
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DOI: https://doi.org/10.1007/s10623-019-00641-w
Keywords
- Permutation codes
- Block permutation metric
- Gilbert–Varshamov bound
- Sphere-packing bound
- Independence number