Quasi‐symmetric designs with fixed difference of block intersection numbers

Abstract

Abstract:The following results for proper quasi-symmetric designs with non-zero intersection numbers x, y and λ > 1 are proved.

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.…”

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

“…Quasi-symmetric designs are called triangle free if it has no three distinct blocks such that any two of them intersect in x points. In [18] it is proved that triangle free quasi-symmetric designs with y = x + 2 do not exist. In [1,[3][4][5][6][7] and [8] several necessary conditions are obtained for the existence of a quasi-symmetric D(v, b, r, k, λ; x, y) design by imposing divisibility restrictions on y − x.…”

“…Quasi-symmetric designs with y = x + 1 are characterized in [18]. Quasi-symmetric designs are called triangle free if it has no three distinct blocks such that any two of them intersect in x points.…”