Difference sets and Hadamard matrices （差集和阿达玛矩阵）.pdf

Difference sets and Hadamard matrices Padraig Ó Catháin National University of Ireland, Galway 14 March 2012 Padraig Ó Catháin Difference sets and Hadamard matrices 14 March 2012 Outline 1 (Finite) Projective planes 2 Symmetric Designs 3 Difference sets 4 Doubly transitive group actions on Hadamard matrices Padraig Ó Catháin Difference sets and Hadamard matrices 14 March 2012 (Finite) Projective planes Projective planes Definition Let V be a set whose elements are points, and let B be a set of subsets of V whose elements are called lines. Then V B is a projective plane if the following axioms hold: 1 Any two distinct points are incident with a unique line. 2 Any two distinct lines are incident with a unique point. 3 There exist four points no three of which are co-linear. Bijections are easily established between: 1 The lines containing x and the points on a line not containing x . 2 The points of two distinct lines. 3 The number of lines and the number of points, etc. Padraig Ó Catháin Difference sets and Hadamard matrices 14 March 2012 (Finite) Projective planes Finite projective planes Let be any field. Then there exists a projective plane over derived from a 3-dimensional -vector space. In the case that is a finite field of order q we obtain a geometry with q2 q 1 points and q2 q 1 lines. q 1 points on every line and q 1 lines through every point. Every pair of lines intersecting in a unique point. Symmetric designs are a generalization of finite projective planes, and give a unified approach to many combinatorial objects. Padraig Ó Catháin Difference sets and Hadamard matrices 14 March 2012 Symmetric Des