Quasi-Symmetric 3-Designs and Elliptic Curves

Abstract

A quasi-symmetric t-design is a t-design with two block intersection sizes p and q (where p < q). Quasi-symmetric 3-designs are classified with p 1. The only nontrivial examples are the 4-(23, 7, Witt design, and its residual, a 3- (22,7,4) design. This proves a conjecture of Sane and Shrikhande. The method is to reduce the classification problem to that of finding all integer points on the elliptic curves y2 X lX + 32X and y2 x 4x + 4.1. Quasi-symmetric designs. A t-(v, k, )) design is a collection 23 of subsets of a v-set (called blocks), such that every block contains k points, and any set of points is in exactly ), blocks. In a t-design, let ); denote the number of blocks containing a given set of points, with 0 < =< t. The identities (1) )k ), i=0,1 restrict the possible parameter sets. A symmetric design is a 2-(v, k, ,) design, such that 0 o, hi k, and any two blocks meet in , points. A t-design with two block intersection sizes is said to be quasi-symmetric. We shall consider quasi-symmetric 3-(v, k, X) designs, with block intersection sizes p and q, where p < q. Calderbank [5] and Sane and Shrikhande [15] proved that the parameters of a quasi-symmetric 2-design 23 satisfy(2) pq(v-1)(v-2)-k(k-1)(v-2)(p+q-1)+k(k-1)2(k-2) >=0, and that equality holds if and only if 23 is a 3-design. Neumaier [13], [14, Prop. 12] had earlier derived an inequality equivalent to (2), but that is not as simple to state. Sane and Shrikhande [15, Conj. 4.6] have conjectured that there exist only twonontrivial, quasi-symmetric 3-designs with p (we consider the designs formed by (v 2)-subsets of a v-set to be trivial). The first example is the 4-(23, 7, Witt design, with q 3, and the second example is its residual, a 3-(22, 7, 4) design (also with q 3). Both these designs are unique. The purpose of this paper is to prove this conjecture.One interesting property of quasi-symmetric 2-designs is that the block graph is strongly regular (see Goethals and Seidel [10], or Cameron and van Lint [7]). The vertices of this graph are the blocks of design, and two blocks are joined if they meet in q points. Recall that a strongly regular graph is regular, and that the number of vertices joined to two given vertices zl and z2 (zl 4 z2), depends only on whether or not Zl and

Some characterizations of quasi-symmetric designs with a spread

“…It was conjectured in [13], that these are the only possibilities. This conjecture was proved true by Calderbank and Morton [4]. See Sane and Pawale [11] for an alternative short proof.…”

“…There has been much recent interest in such designs. For example, for t = 2: [1], [3], [9], [10] and in case t = 3: [4], [11], [13].…”

Nearly triply regular Drad’s ofRH type

“…Now in recent years, quasi-symmetric 2-designs were worked out extensively and deep results were obtained. For example, see [1,9,10]. However, it seems to us that the main stream of the development scarcely touches the type of quasi-symmetric 2-designs mentioned above, although, on the other hand, basic tools for investigating nearly triply regular symmetric 2-designs so far can be said to be coming from quasi-symmetric 2-designs.…”

“…Equation (1) shows that the valency of F equals k and that for any two distinct vertices of F there exist exactly 2 vertices in the intersections of their out-and in-neighborhoods respectively. Moreover, since A + A ~ is a (0, 1) matrix, F is asymmetric.…”

“…AA t = (k -2)1 + 2J = A~A (1) where I and J are the identity and all one matrices of size v respectively. Equation (1) shows that the valency of F equals k and that for any two distinct vertices of F there exist exactly 2 vertices in the intersections of their out-and in-neighborhoods respectively.…”