1 / 11

Definitions

Definitions. Let i) standard q-ary alphabet. ii) is the set of all q! permutations of q symbols. is a set of n elements. n -sequence. q -partition. For any. -complement. v). Number of positions i , i=1,2,…n where:.

marianne-mercier
Télécharger la présentation

Definitions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Definitions Let i) standard q-ary alphabet. ii) is the set of all q! permutations of q symbols. • is a set of n elements n-sequence q-partition

  2. For any -complement v) Number of positions i, i=1,2,…n where: vi) Hamming Distance: • vii) Partition Distance

  3. Bound on partition distance:

  4. viii) The q-ary ( ) matrix:

  5. Applications • An individual is asked to split n objects up into parts, putting • seemingly similar objects into the same part. • On the base of preliminary testing of individuals with K known diagnoses one • can choose from a subset of q-partitions • corresponding to the K putative subtypes of the complex disease. • The patient’s result will be some partition . We then calculate the partition distance between and partitions in the K set of putative sub-types. The subtype(s) that are represented by the partitions that yield the smallest distance is possibly what the individual is suffering from.

  6. Varshamov-Gilbert and Hamming bounds. Spheres The sphere centered at x of radius tis the set of codewords: The volume of any sphere (not in a partition code) of radius t is: Hamming Bound Suppose we have a code with minimum distance d=2t+1. Let M denote the maximal possible size of the code. Then around each codeword we will have a sphere of radius t. Since all spheres are disjoint then we must have:

  7. Varshamov-Gilbert Bound

  8. STIRLING SET NUMBERS By definition a Stirling set number (Stirling number of the second kind is the number of ways of partitioning a set of n elements intok non-empty clusters. Denoted by: All possible q-partitions of an n set:

  9. SPHERE VOLUME (PARTITION CODE) Space of un-ordered q-partitions endowed with partition distance is homogeneous. Partitions of distance k from 0 partition: If we have i>k then S(k,i)=0 Volume of Sphere:

  10. Bounds for q-partition codes Hamming Bound Let M denote the maximal possible size of a q-partition code with, length n and minimum distance d= 2t+1 . Then we have that the Hamming bound is:

  11. Varshamov-Gilbert Bound:

More Related