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== Bounds ==
We can look at a few different lower bounds on the permutation codes. Using the results from the theory of MDS, or Maximum Distance Separable, codes to provide two new lower bounds. The first of the two is when you are dealing with the next prime power larger than or equal to n is smaller than the next prime larger than or equal to n. The second has to do with large distances. We use the results of a theorem to be able to understand further theorems that get us to these MDS lower bounds.
'''Theorem 1:''' Let d,k,n be integers such that 0 < k < n and 1 < d ≤ n. Moreover, let q be a prime power and s,r be positive integers such that n = qs+r and 0 ≤ r < q. If there exists an [n,k,d]q code C such that C^⊥ has a codeword of Hamming weight n, then
M(n,d) ≥ {n!M(K,d)}/{(s + 1)!^rs!^{q-r} q^{n-k-1}},
whereK=(S_{s+1})^r ×(S_s)^{q-r} .
The proof is extensive and can be found here <ref>https://www.adarshsrinivasan.com/publication/permutation-error-correcting-codes-and-their-applications-to-public-key-cryptography/permutation-error-correcting-codes-and-their-applications-to-public-key-cryptography.pdf.
</ref>.
Now we can take a look at another theorem that follows directly from the previous.
'''Theorem 2:'''For every prime power q ≥ n, and every integer d with 2<d<n,
M(nod) ≥ n!/q^{d-2}.
The proof of this follows directly from the previous theorem. This second theorem provides a lower bound on M(nod) using the existence of MDS codes of length n over a finite field with cardinality at least n.
Now we can take a look into the lower bounds that deal with large distances. Another theorem can be obtained from the first, and it states
'''Theorem 3:'''For every prime power q, and every 3 < d < q,
M(q+1,d) ≥ (q+1)^!/2q^{d-2}.
Again, the proof follows directly from the first theorem. With the second theorem above this allows our q to be the next prime power greater than or equal to n, we beat - or at least equal - previously found bounds. If n-1 is the next prime power, the third theorem shown above beats previously found bounds asymptotically in large distance.
== References ==
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