The Existence of a Bush-Type Hadamard Matrix of Order 324 and Two New Infinite Classes of Symmetric Designs

Abstract

A symmetric 2-(324, 153, 72) design is constructed that admits a tactical decomposition into 18 point and block classes of size 18 such that every point is in either 0 or 9 blocks from a given block class, and every block contains either 0 or 9 points from a given point class. This design is self-dual and yields a symmetric Hadamard matrix of order 324 of Bush type, being the first known example of a symmetric Bush-type Hadamard matrix of order 4n^2 for n > 1 odd. Equivalently, the design yields a strongly regular graph with parameters v=324, k=153, \lambda=\mu=72 that admits a spread of cocliques of size 18. The Bush-type Hadamard matrix of order 324 leads to two new infinite classes of symmetric designs with parameters v=324(289^m+289^{m-1}+\cdots+289+1), \quad k=153(289)^m, \quad \lambda=72(289)^m, and v=324(361^m+361^{m-1}+\cdots+361+1), \quad k=171(361)^m, \quad \lambda=90(361)^m, where m is an arbitrary positive integer.