Genetics has enabled us to detect a class of diseases known as complex diseases. At the present time, diagnosis of complex diseases such as ADHD is a problem that is still being studied. The mathematical tools that we discuss will aid the analysis of complex disease data. These methods present new implications of the partitioning of data sequences. We define a new concept of distance (based on the Hamming distance) between two distinct sets of unordered partitions that we call a partition-distance. We can verify that this distance is a valid metric in the space of unordered partitions of any finite set S size n, where each partition contains
disjoint subsets of S. Using
the distinct partitions of a set S, endowed with the proposed
metric, we investigate a new class of codes which we call
q-partition codes. These codes will aid
the study currently being held at Los-Alamos National Laboratory and
UC Irvine on the deconvolution of complex disease
data. Particular applications of these codes will be discussed. An
example of a 3-partition code based on conventional linear codes
endowed with the partition-distance as well as the bounds on the rate
of q-partition codes will be considered. Finally, a construction of
q-partition codes based on first order Reed-Muller codes will be
examined. The project is a collaboration between UC Irvine, Los Alamos
National Laboratory, UNO, Moscow State University and MIT.
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