Project Title: Partition Codes: Analyses and Generation.
Adviser : Vyacheslav Rykov
Description :
We will study two distance concepts between q-nary n-sequences, called partition distances. These distances are metrics in the space of all partitions of a finite n-set. For the metrics, we study codes called q-partition codes and find constructions of these codes. A random coding bound will be obtained on the base of calculating an average volume of the spheres in this metric space. We also will work out an application of q-partition codes to the statistical analysis of psychological or medical tests using questionnaires.
A . Study the theoretical aspects of the problem by using following sources:
1. A D'yachkov, V Rykov, D. Torney, S. Yekhanin, “On Application of the partition distance concept to a comparative analysis of psychological or sociological tests”, Stochastic Analysis and Applications , 2005
2. A.G.D'yachkov,
V.V.Rykov, D.C.Torney, S.M.Yekhanin, “On codes for partition distance”, 9th International Workshop " Algebraic and Combinatorial Coding Theory "
,
3. A.G.D'yachkov, V.V.Rykov, D.C.Torney, S.M.Yekhanin, “Partition Codes”. 2004 IEEE International Symposium on Information Theory , Chicago, USA, June 27 - July 1, 2004, p.256.
4. D.Gusfield, “Partition-distance: A problem and class of perfect graphs arising in clustering”, Information Processing Letters , 82,(2002), pp.159-164.
5. B.G. Mirkin, L.B. Tcherny, “On measurement of proximity between various partitions of finite set”, Avtomatika i Telemekhanika , 1970, No. 5, pp.120-127.
6. A. Serfling, “Approximation Theorems of Mathematical Statistics” , John Wiley, 1985.
7. F.J. MacWilliams,
N.J.A. Sloan, “The Theory of Error-Correcting Codes”,
This will help her/him understand the research topic and serve as an introduction to the final, written, research report.
B. Develop algorithms , generating q-nary strings.
C . Write computer programs for calculating partition distance and generating partition codes.
D. Run the programs and help the adviser in statistical analyses of the partition distance functions and generating partition codes.
E. Put together her/his findings, written software, significant graphs, tables, and so on in the final research report to be presented at the MAM Symposium.
OTHER REQUIREMENTS: The students interested in the project above are expected to have taken and passed with maximal grades or close MATH 1950(Calc I), MATH 1960(Calc II), MATH 4050 (Linear Algebra) and MATH 4740 (Introduction to Probability and Statistics) . They should be familiar with computers and MAPLE. The student is also expected to meet with the adviser a couple of times a week, (or communicate actively by e-mail) for discussions, guidance, and progress reports during the preparation period of the project.