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== Gilbert-Varshamov Bound ==
A main problem in permutation codes is to determine the value of <math>M(n,d)</math>, where <math>M(n,d)</math> is defined to be the maximum number of codewords in such a code. There has been little progress made for <math>4 \leq d \leq n-1</math>, except for small lengths. We can define <math>D(n,k)</math> with <math>k \in \{0, 1, ..., n\}</math> to denote the set of all permutations in <math>S_n</math> which have exactly <math>k</math> distance from the identity.
<math>D(n,k)= \{ \sigma \in S_n: d_H (\sigma, id)=k\}</math> with <math>|D(n,k)|={n choose k}D_k</math>. where <math>D_k</math> is the number of [[Derangement|derangements]] of order <math>k</math>.
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