In this paper we show that Lander's coding-theoretic proof of (parts of) the Bruck-Ryser-Chowla Theorem can be suitably modified to obtain analogous number theoretic restrictions on the parameters of quasi-symmetric designs. These results may be thought of as extensions to odd primes of the recent binary nonexistence results due to Calderbank et al. The results in this paper kill infinitely many feasible parameters for quasi-symmetric designs.
Recall [6], [16], that a 2 -(v, k, ~,) design is an incidence system on v points with k points on each block and ~ blocks through each pair of points. The total number of blocks and the number of blocks through each point in such a design are usually denoted by b and r respectively. These ancillary parameters are related to the main parameters by: bk = rv, r(k -1) = X(v -1).
(1)
To display the ancillary parameters along with the main parameters, such a design is also referred to as a (b, v, r, k, ~) design.
A 2 -(v, k, )x) design is said to be quasi-symmetric if there are numbers ct ;~ B such that any two distinct blocks have ct or/3 points in common, and both numbers actually occur. To avoid trivialities, we assume or, ~ < k. The numbers ct, B are called the intersection numbers of the design. Recall [6] that a strongly regular graph with parameters (n, a, c, d) is a regular graph of degree a on n vertices such that any two vertices have c or d common neighbors according as these two vertices are adjacent or not. The block graph I TM of a quasi-symmetric design is the graph with its blocks as vertices and intersection in ctpoints as adjacency. (Note that there is an ambiguity here since interchanging t~,/3 replaces F by its complement. This ambiguity is usually removed by requiring that ~ > /3. In this paper we prefer to retain the symmetry between the intersection numbers at the cost of this little ambiguity: the block graph is defined only up to complementation.) F is a strongly regular graph (see [1] and [11]). Its eigenvalues are (rk -bB)/(u -B) -(k -f3)/(~ -13) with multiplicity 1, (r -X)/(ct -B) -(k -B)/(ol -/3) with multiplicity v -1 and -(k -B)/(c~ -B) with multiplicity b -v. Being rational numbers as well as algebraic integers, these eigenvalues must be rational integers. Thus we have the following well-known divisibility conditions on the parameters of a quasi-symmetric design:
-131r -X, o~ -/3lk -t3, c~ -/3l(b -;~)t3.
(2)