On Strong Tactical Decompositions

If a 2-(v, k, k) design 2) admits a tactical decomposition with d point classes and c block classes then b — v ^ c—d ^ 0, (see [3]). Decompositions for which b+d = v + c are of special interest (see for instance [1]), and are called strong. Any tactical decomposition of a symmetric design is strong. A strong tactical decomposition of a design is called a strong resolution if it has only one point class (and the design is called strongly resolvable). Any affine design is strongly resolvable with the parallel classes giving the block classes of the strong resolution. Shrikhande and Raghavarao [7] constructed a family of designs admitting a strong resolution, which are neither symmetric nor affine. A design admitting a strong resolution can be shown to have at most two intersection numbers (see [5]). (The intersection numbers of a design are the non-diagonal entries of AA, where A is an incidence matrix of the design). In [2] it was shown that whenever there exist affine planes of orders n — 1 and n, then there exists a 2-((« + l)(«— I), n{n — 1),«) design admitting a strong tactical decomposition. These designs were shown to have exactly three intersection numbers, and hence would never admit a strong resolution. In fact all known examples of designs admitting strong tactical decompositions have at most three intersection numbers. Thus it appears that if a design admits a strong tactical decomposition this imposes powerful restrictions on the intersection numbers of the design. The purpose of this paper will be to prove the following theorem.