Permutation code: Difference between revisions

Revision as of 13:57, 24 October 2022

Revision as of 18:47, 25 October 2022

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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 [[Power-line communication|powerline]] communication less impactful on codewords

== BoundsUpper Bound ==

Gilbert-Varshamov (page 2 of the document)

==Lower Bound==

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.

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'''Theorem 2:'''For every prime power q ≥ n, and every integer d with 2<d<n,

M(nodn,d) ≥ 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.