Infinite families of non-embeddable quasi-residual Menon designs

The notion of residual and derived design of a symmetric design was introduced in a classic paper by R. C. Bose (1939). A quasi-residual (quasi-derived) design is a 2-design which has the parameters of a residual (derived) design. The embedding problem of a quasi-residual design into a symmetric design is an old and natural question. A Menon design of order h² is a symmetric (4h²,2h²-h, h²-h) design. Quasi-residual and quasi-derived designs of a Menon design have parameters 2-(2h²+h,h²,h²-h) and 2-(2h²-h,h²-h,h²-h-1), respectively.

We use regular Hadamard matrices to construct non-embeddable quasi-residual and quasi-derived Menon designs. As applications, the first two new infinite families of non-embeddable quasi-residual and quasi-derived Menon designs are constructed. This is a joint work with T. A. Alraqad.