New quasi-symmetric designs by the Kramer–Mesner method

A tt-(v,k,λ)(v,k,λ) design is quasi-symmetric if there are only two block intersection sizes. We adapt the Kramer–Mesner construction method for designs with prescribed automorphism groups to the quasi-symmetric case. Using the adapted method, we find many new quasi-symmetric 22-(28,12,11)(28,12,11) and 22-(36,16,12)(36,16,12) designs, establish the existence of quasi-symmetric 22-(56,16,18)(56,16,18) designs, and find three new unitals 22-(217,7,1)(217,7,1) of non-prime power order.