AbstractA 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 4n2 for n > 1 odd. Equivalently, the design yields a strongly regular graph with parameters v=324, k=153, λ=μ=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
$$v = 324(289^m + 289^{m - 1} + \cdot \cdot \cdot + 289 + 1),{\text{ }}k = 153(289)^m ,{\text{ }}\lambda \;{\text{ = }}\;{\text{72(289)}}^m ,$$
and
$$v = 324(361^m + 361^{m - 1} + \cdot \cdot \cdot + 361 + 1),{\text{ }}k = 171(361)^m ,{\text{ }}\lambda \;{\text{ = }}\;90{\text{(361)}}^m ,$$
where m is an arbitrary positive integer.
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