A note on quasi-symmetric designs

A quasi-symmetric design is a (v, k, λ) design with two intersection numbers x, y where 0 ≤ x < y < k. We show that for fixed x, y, λ with x > 1, λ > 1, y = λ and λ (4xy + ((y − x) − 2x− 2y + 1)λ) a perfect square of a positive integer, there exist finitely many quasi-symmetric designs. We rule out the possibilities of quasi-symmetric designs corresponding to y = x + 3 and (λ, x) = (9, 2), (8, 3) and prove that the only feasible parameters associated with x = 2, y = 5, λ = 8 and x = 4, y = 7, λ = 9 are v = 1001, k = 65 and v = 4642, k = 154 respectively. These are some of the open cases reported in the classification of quasi-symmetric designs with y = x + 3, x > 0 obtained by Mavron, McDonough and Shrikhande [Des. Codes Crypto. 63 (2012) no. 1, 73–86]. The technique used rules out the possibilities of quasi-symmetric designs corresponding to x = 1 and λ = (y − 1) for y ≥ 4.