On Two Classes of Menon Designs

Let p and 2 p − 1 be prime powers and p ≡ 3 ( mod 4 ) . Then there exists a symmetric design with parameters ( 4 p 2 , 2 p 2 − p , p 2 − p ) . Thus there exists a regular Hadamard matrix of order 4 p 2 . In a similar way we construct a symmetric ( 4 ( p + 1 ) 2 , 2 p 2 + 3 p + 1 , p 2 + p ) design D when p and 2 p + 3 are prime powers, such that p ≡ 3 ( mod 4 ) .