Permutation code

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 ${\displaystyle C}$  is defined as a subset of the Symmetric Group in ${\displaystyle S_{n}}$  endowed with the usual Hamming distance between strings of length ${\displaystyle n}$ . More precisely, if ${\displaystyle \sigma ,\tau }$  are permutations in ${\displaystyle S_{n}}$ , then${\displaystyle d(\tau ,\sigma )=|\left\{i\in \{1,2,...,n\}:\sigma (i)\neq \tau (i)\right\}|}$ 

The minimum distance of a permutation code is defined to be the minimum positive integer ${\displaystyle d_{min}}$  such that there exist ${\displaystyle \sigma ,\tau }$  ${\displaystyle \in }$  ${\displaystyle C}$ , distinct, such that ${\displaystyle d(\sigma ,\tau )=d_{min}}$ .

One of the reasons why permutation codes are suitable for certain channels is that alphabet symbols only appear once in each codeword, which for example makes the errors occurring in the context of powerline communication less impactful on codewords

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.