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Level: BSc, MSci
Title: T-designs
Supervisor:

Prof. Leonard Soicher
Research Area: Combinatorics
Description:

A block design is a pair (V,B), such that V is a finite non-empty set and B is a finite collection of non-empty subsets of V, called blocks, such that every element of V is in at least one block. A t-design is a special type of block design. More precisely, for t a non-negative integer, a t-design, or more specifically a t-(v,k,λ) design, is a block design (V,B), such that V has exactly v elements, each block has the same size k, and each t-element subset of V is contained in the same positive number λ of blocks. For example, if V = {1,...,7} and B = {1,2,4}, {2,3,5}, {3,4,6}, {4,5,7}, {1,5,6}, {2,6,7}, {1,3,7}, then (V,B) is a 2-(7,3,1) design (it is also a 1-(7,3,3) design and a 0-(7,3,7) design).
There are applications of t-designs to many areas of mathematics, including the design of experiments, group theory, coding theory and finite geometry. This project would include the basic theory of t-designs, followed either by a selected application, or one or more significant results (with proof) in the theory of t-designs.
Further Reading:

Literature suggestions will be supplied.
Key Modules:

MTH6109 Combinatorics (desirable)
Other Information:
Current Availability: Yes