Quasi-symmetric designs with the difference of block intersection numbers two

Abstract

Quasi-symmetric designs with intersection numbers x > 0 and y = x + 2 under the condition λ > 1 are investigated. If D (v, b, r, k, λ; x, y) is a quasi-symmetric design with above conditions then it is shown that either λ = x + 1 or x + 2 or D is a design with the parameters given in the Table 6 or complement of one of these designs.

On some parametric classifications of quasi-symmetric 2-designs

“…Let D be a quasi-symmetric 2-design with standard parameter set (2,18,2496), (3,7,1456), (3,11,3200), (4,12,3840), (4,19,2964), (5,13,4544), (6,14,5312), (7,15,6144), (9,24,2841). But for these values of x, k and M , we are not getting any positive integral solutions of λ from the quadratic equation Aλ 2 + B λ + C = 0 and hence no triangle free quasi-symmetric 2-designs can exist under these parametrical restrictions.…”

“…In [5], several necessary conditions are obtained for the existence of a quasisymmetric 2-design with parameter set D (v, b, r, k, λ; x, y) by imposing the divisibility restrictions on y − x. It was also shown in [16] that there are finitely many such designs for y ≥ 2 and fixed block size k. Most of the recent works on quasi-symmetric 2-designs have been concentrated on the difference of the intersection numbers x and y, where y = x + 2 in [14] and [10] i.e. the difference of the intersection numbers is 2 and 3.…”

On quasi-symmetric designs with intersection difference three

“…So quasi-symmetric designs whose block graph and its complement are connected give rise to primitive strongly regular graphs. So earlier papers, such as [1,11,[18][19][20][21][22]25], and [23] may be viewed under this wider umbrella.…”

“…In a recent preprint, Pawale [22] obtained a parametric classification of proper quasisymmetric 2-designs with y − x = 2, with x > 0 and λ > 1. It is shown in [22] that if D is a quasi-symmetric 2-design with these conditions, then either λ = x + 1 or λ = x + 2, or D is a design with parameters given in the form of an explicit table, or the complement of one of these designs.…”

“…It is shown in [22] that if D is a quasi-symmetric 2-design with these conditions, then either λ = x + 1 or λ = x + 2, or D is a design with parameters given in the form of an explicit table, or the complement of one of these designs.…”