On symmetric and quasi-symmetric designs with the symmetric difference property and their codes

AbstractThe four symmetric 2-(64, 28, 12) designs with the symmetric difference property are characterized as the only designs with the given parameters and minimal rank over GF(2). These designs give non-isomorphic quasi-symmetric 2-(36, 16, 12) and 2-(28, 12, 11) designs as residual and derived designs. The binary codes of the quasi-symmetric 2-(28, 12, 11) designs provide four inequivalent self-orthogonal doubly-even (28, 7, 12) codes. This gives a negative answer to the question for the uniqueness of the code of the Hermitian unital of order 3