Permutation code

Permutation codes are a family of error correction codes 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 ${\displaystyle C}$  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 the 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

Gilbert-Varshamov Bound

A main problem in permutation codes is to determine the value of ${\displaystyle M(n,d)}$ , where ${\displaystyle M(n,d)}$  is defined to be the maximum number of codewords in a permutation code of length ${\displaystyle n}$  and minimum distance ${\displaystyle d}$ . There has been little progress made for ${\displaystyle 4\leq d\leq n-1}$ , except for small lengths. We can define ${\displaystyle D(n,k)}$  with ${\displaystyle k\in \{0,1,...,n\}}$  to denote the set of all permutations in ${\displaystyle S_{n}}$  which have distance exactly ${\displaystyle k}$  from the identity.

Let ${\displaystyle D(n,k)=\{\sigma \in S_{n}:d_{H}(\sigma ,id)=k\}}$  with ${\displaystyle |D(n,k)|={\tbinom {n}{k}}D_{k}}$ , where ${\displaystyle D_{k}}$  is the number of derangements of order ${\displaystyle k}$ .

The Gilbert-Varshamov bound is a very well known upper bound,^[7] and so far outperforms other bounds for small values of ${\displaystyle d}$ .

Theorem 1: ${\displaystyle {\frac {n!}{\sum _{k=0}^{d-1}|D(n,k)|}}\leq M(n,d)\leq {\frac {n!}{\sum _{k=0}^{[{\frac {d-1}{2}}]}|D(n,k)|}}}$ 

There has been improvements on it for the case where ${\displaystyle d=4}$  ^[7] as the next theorem shows.

Theorem 2: If ${\displaystyle k^{2}\leq n\leq k^{2}+k-2}$  for some integer ${\displaystyle k\geq 2}$ , then

${\displaystyle {\frac {n!}{M(n,4)}}\geq 1+{\frac {(n+1)n(n-1)}{n(n-1)-(n-k^{2})((k+1)^{2}-n)((k+2)(k-1)-n)}}}$ .

For small values of ${\displaystyle n}$  and ${\displaystyle d}$ , researchers have developed various computer searching strategies to directly look for permutation codes with some prescribed automorphisms ^[8]

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. S2CID 124937273.
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. S2CID 214381288.
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. S2CID 18529905.
7. Gao, Fei; Yang, Yiting; Ge, Gennian (May 2013). "An Improvement on the Gilbert–Varshamov Bound for Permutation Codes". IEEE Transactions on Information Theory. 59 (5): 3059–3063. doi:10.1109/tit.2013.2237945. ISSN 0018-9448. S2CID 13397633.
8. Smith, Derek H.; Montemanni, Roberto (2011-08-19). "A new table of permutation codes". Designs, Codes and Cryptography. 63 (2): 241–253. doi:10.1007/s10623-011-9551-8. ISSN 0925-1022. S2CID 207115236.