Bush‐type Hadamard matrices and symmetric designs

Abstact: A symmetric 2‐(100, 45, 20) design is constructed that admits a tactical decomposition into 10 point and block classes of size 10 such that every point is in either 0 or 5 blocks from a given block class, and every block contains either 0 or 5 points from a given point class. This design yields a Bush‐type Hadamard matrix of order 100 that leads to two new infinite classes of symmetric designs with parameters $\nu = 100(81^m+81^{m-1}+\cdots +81+1), \quad \kappa = 45(81)^m, \quad \lambda = 20(81)^m,$ and $\nu = 100(121^m+121^{m-1}+\cdots +121+1), \quad \kappa = 55(121)^m, \quad \lambda = 30(121)^m,$ where m is an arbitrary positive integer. Similarly, a Bush‐type Hadamard matrix of order 36 is constructed and used for the construction of an infinite family of designs with parameters ${v} = 36(25^m + 25^{m-1} + \cdots + 51 + 1),\quad k = 15(25)^m,\quad\lambda = 6(25)^m,$ and a second infinite family of designs with parameters ${v} = 36(49^m + 49^{m-1} + \cdots + 49 +1,\quad k = 21(49)^m,\quad\lambda = 12(49)^m,$ where m is any positive integer. © 2000 John Wiley & Sons, Inc. J Combin Designs 9: 72–78, 2001