Non Existence of Triangle Free Quasi-symmetric Designs

Abstract

The following two results are proved.(1) Let D be a triangle free quasi-symmetric design with k = 2y − x and x ≥ 1 then D is a trivial design with v = 5 and k = 3.(2) There do no exist triangle free quasi-symmetric designs with x ≥ 1 and λ = y or λ = y − 1.

Quasi‐symmetric designs with fixed difference of block intersection numbers

“…Triangle free quasisymmetric 3-designs are completely classified in Pawale [9]. Recently in [10], Pawale characterize triangle free quasi-symmetric designs with non-zero intersection numbers and k = 2y − x, with trivial design. Also it is proved that triangle free quasi-symmetric designs with λ = y or λ = y − 1 do not exist.…”

On some parametric classifications of quasi-symmetric 2-designs

“…Some works have been done on trianglefree quasi-symmetric 2-designs for the intersection numbers 0 and y in [8]. Later, many works have been developed in [10], [12] and [13]. We present here some of the relevant results.…”

“…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.…”

“…During the last several decades quasi-symmetric 2-designs and their classification play an important role to the study of design theory see for example [6,10,12,13,15,16,18]. Many results have been developed in the theory of binary codes, basically on self-complementary codes, self-dual codes using such designs.…”

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.…”