Algebraic Studies of Combinatorial Objects Ming Ming Tan - - PowerPoint PPT Presentation

Algebraic Studies of Combinatorial Objects

Ming Ming Tan Supervisor: Bernhard Schmidt

Nanyang Technological University, Singapore

January 19, 2015

1 / 5

Combinatorial Objects

Hadamard Matrices       1 1 1 1 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1       (2m, 2, 2m, m) Relative Difference Sets Hadamard Groups Cocyclic Hadamard Matrices Group Invariant Hadamard Matrices       1 1 1 −1 −1 1 1 1 1 −1 1 1 1 1 −1 1       Group Invariant Weighing Matrices two-weight irreducible cyclic codes planar functions

2 / 5

Algebraic Connections: Examples

Circulant Hadamard Matrices M =       1 1 1 −1 −1 1 1 1 1 −1 1 1 1 1 −1 1       MMT = 4I X = 1 + ζ + ζ2 − ζ3 |X|2 = 4 (2, 2, 2, 1) Relative Difference Sets Z4 = {0, 1, 2, 3} N = {0, 2} R = {0, 1} △R = {r − r′ : r, r′ ∈ R, r = r′} = Z4 \ N X = 1 + ζ |X|2 = 2

3 / 5

New Methods & Results

Circulant Weighing Matrices Non-existence of infinite families

◮ Generalized multiplier

theorem

◮ Weil number ◮ Field descent

(2m, 2, 2m, m) Relative Difference Sets Existence of infinite families

◮ Golay sequences (binary

and quaternary)

◮ Williamson matrices ◮ Hadamard difference sets

4 / 5

Questions

◮ Can these new methods be applied to other combinatorial

• bjects?

◮ What are the connections of these combinatorial objects with

codes and planar functions?

5 / 5