User:Brightdan/sandbox

= Permutation Codes =

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

Definition and Properties
A permutation code $$C$$ can be defined as a subset of Symmetric Group in $$S_n$$ endowed with the usual Hamming distance between strings of length $$n$$. More precisely, if $$\sigma, \tau$$ are the permutation in $$S_n$$, then d($$\sigma, \tau$$) = $$|\left \{ i \in \{1, 2, ..., n\} : \sigma(i) \neq \tau(i) \right \}|$$

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

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