The existence of a Bush-type Hadamard matrix of order 324 and two new infinite classes of symmetric designs

A symmetric 2-(324, 153, 72) design is constructed that admits a tactical d ecomposition into 18 point and block classes of size 18 such that every poi nt is in either 0 or 9 blocks from a given block class, and every block con tains either 0 or 9 points from a given point class. This design is self-du al 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 4 n(2) for n > 1 odd. Equivalently, the design yields a strongly regular grap h with parameters v = 324, k = 153, lambda = mu = 72 that admits a spread o f 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) +...+ 289 + 1), k = 153(289)(m), lambda = 72(289) (m), and v = 324(361(m) + 361(m-1) +...+ 361 + 1), k = 171(361)(m), lambda = 90(361) (m), where in is an arbitrary positive integer.