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 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(289m+289m−1+⋅⋅⋅+289+1), k=153(289)m, λ = 72(289)m, and v=324(361m+361m−1+⋅⋅⋅+361+1), k=171(361)m, λ = 90(361)m, where m is an arbitrary positive integer.
Designs, Codes and Cryptography
The existence of a Bush-Type Hadamard Matrix of order 324 and two new infinite classes of symmetric designs.
Designs, Codes and Cryptography,
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