On quasi-symmetric designs with intersection difference three

In a recent paper, Pawale (Des Codes Cryptogr, 2010) investigated quasi-symmetric 2-(v, k, λ) designs with intersection numbers x > 0 and y = x + 2 with λ > 1 and showed that under these conditions either λ = x + 1 or λ = x + 2, or D{mathcal{D}} is a design with parameters given in the form of an explicit table, or the complement of one of these designs. In this paper, quasi-symmetric designs with y − x = 3 are investigated. It is shown that such a design or its complement has parameter set which is one of finitely many which are listed explicitly or λ ≤ x + 4 or 0 ≤ x ≤ 1 or the pair (λ, x) is one of (7, 2), (8, 2), (9, 2), (10, 2), (8, 3), (9, 3), (9, 4) and (10, 5). It is also shown that there are no triangle-free quasi-symmetric designs with positive intersection numbers x and y with y = x + 3.