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 Group Invariant Hadamard Matrices   1 1 1 1     1 1 − 1 − 1 1 1 1 − 1     1 − 1 1 − 1  − 1 1 1 1          1 − 1 − 1 1 1 − 1 1 1     1 1 − 1 1 (2 m, 2 , 2 m, m ) Group Invariant Relative Difference Sets Weighing Matrices Hadamard Groups two-weight irreducible cyclic codes Cocyclic Hadamard Matrices planar functions 2 / 5

Algebraic Connections: Examples Circulant Hadamard (2 , 2 , 2 , 1) Relative Difference Matrices Sets Z 4 = { 0 , 1 , 2 , 3 }   1 1 1 − 1 N = { 0 , 2 }   − 1 1 1 1   M = R = { 0 , 1 }   1 − 1 1 1     1 1 − 1 1 △ R = { r − r ′ : r, r ′ ∈ R, r � = r ′ } MM T = 4 I = Z 4 \ N X = 1 + ζ + ζ 2 − ζ 3 X = 1 + ζ | X | 2 = 4 | X | 2 = 2 3 / 5

New Methods & Results Circulant Weighing (2 m, 2 , 2 m, m ) Relative Matrices Difference Sets Non-existence of infinite Existence of infinite families families ◮ Golay sequences (binary ◮ Generalized multiplier and quaternary) theorem ◮ Williamson matrices ◮ Weil number ◮ Hadamard difference sets ◮ Field descent 4 / 5

Questions ◮ Can these new methods be applied to other combinatorial objects? ◮ What are the connections of these combinatorial objects with codes and planar functions? 5 / 5