Permutation code: Difference between revisions

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=== Lower bounds using linear codes ===
Using a <math>[n, k, d]_q</math> linear block code, one can prove that there exists a permutation code in the symmetric group of degree <math>n</math>, having minimum distance at least <math>d</math> and large cardinality.<ref name=":1">G. Micheli and A. Neri, "New Lower Bounds for Permutation Codes Using Linear Block Codes," in IEEE Transactions on Information Theory, vol. 66, no. 7, pp. 4019-4025, July 2020, doi: 10.1109/TIT.2019.2957354.</ref>. A lower bound for permutation codes that provides asymptotic improvements in certain regimes of length and distance of the permutation code<ref name=":1" /> is discussed below. For a given subset <math>\Kappa</math> of the symmetric group <math>S_n</math>, we denote by <math>M(\Kappa, d)</math> the maximum cardinality of a permutation code of minimum distance at least <math>d</math> entirely contained in <math>\Kappa</math>, i.e.
 
<math>M(\Kappa, d) = max\{|\Gamma| : \Gamma \subset \Kappa , d(\Gamma) \ge d\}</math>.