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

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

Bound on partition distance:

viii) The q-ary ( ) matrix:

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.

Varshamov-Gilbert and Hamming bounds. Spheres The sphere centered at x of radius t is 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:

Varshamov-Gilbert Bound

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 into k non-empty clusters. Denoted by: All possible q-partitions of an n set:

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:

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:

Varshamov-Gilbert Bound: