Permutation code

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Permutation codes are a family of error correction code that were introduced first by Slepian in 1965 [1] [2] and have been widely studied both in Combinatorics [3][4] and Information theory due to their applications related to Flash memory [5] and Power-line communication [6].

Definition and Properties

A permutation code can be defined as a subset of Permutation in   endowed with the usual Hamming distance between strings of length  . More precisely, if   are the permutation in  , then d( )  ,

The minimum distance of a permutation code is defined to be the minimum positive integer   such that there exist       such that d( ) =  .

Bounds

TODO

References

  1. ^ "Codes on Euclidean Spheres, Volume 63 - 1st Edition". www.elsevier.com. Retrieved 2022-09-20. Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 2001.
  2. ^ Slepian, D. (March 1965). "Permutation modulation". Proceedings of the IEEE. 53 (3): 228–236. doi:10.1109/PROC.1965.3680. ISSN 1558-2256.
  3. ^ Cameron, Peter J. (2010-02-01). "Permutation codes". European Journal of Combinatorics. 31 (2): 482–490. doi:10.1016/j.ejc.2009.03.044. ISSN 0195-6698.
  4. ^ Tarnanen, H. (January 1999). "Upper Bounds on Permutation Codes via Linear Programming". European Journal of Combinatorics. 20 (1): 101–114. doi:10.1006/eujc.1998.0272. ISSN 0195-6698. J. Combin., 20(1):101–114, 1999
  5. ^ Han, Hui; Mu, Jianjun; He, Yu-Cheng; Jiao, Xiaopeng; Ma, Wenping (April 2020). "Multi-Permutation Codes Correcting a Single Burst Unstable Deletions in Flash Memory". IEEE Communications Letters. 24 (4): 720–724. doi:10.1109/LCOMM.2020.2966619. ISSN 1089-7798.
  6. ^ Chu, Wensong; Colbourn, Charles J.; Dukes, Peter (May 2004). "Constructions for Permutation Codes in Powerline Communications". Designs, Codes and Cryptography. 32 (1–3): 51–64. doi:10.1023/b:desi.0000029212.52214.71. ISSN 0925-1022.