Block transitive resolutions of t-designs and room rectangles

✍ Scribed by Robert A. Liebler; Spyros S. Magliveras; Sergei V. Tsaranov

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 779 KB
• Volume: 58
• Category: Article
• ISSN: 0378-3758
• DOI: 10.1016/s0378-3758(96)00065-1

✦ Synopsis

By a resolution of t-designs we mean a partition of the trivial design (x) of all k-subsets of a v-set X into t -(v',k, 2) designs, where vt<~v. A resolution of t-designs with v = v ~ is also called a large set of t-designs. A Room rectangle R, based on (x), is a rectangular array whose non-empty entries are k-sets. This array has the further property that taken together the rows form a resolution of tl-designs, and the columns form a resolution of t2-designs. A resolution of t-designs for (x) is said to admit G as a block transitive automorphism group if G is k-homogeneous on X, and permutes the t-designs of the resolution among themselves.

Some examples of block transitive resolutions of nontrivial t-designs, for t~>2, are: (1) an Mll-invariant set of 3-(10, 4, 1) designs, (2) an Ml2-invariant set of 4-(11, 5, 1) designs, (3) an M24-invariant set of 2-(21, 5, 1) designs, (4) a PFL2(2s)-invariant set of 3-(2 *, 4, 1) designs (s = 3 or 5), (5) a PFL2(32)-invariant set of 2-(16,4, 1) designs, and (6) a variety of PSL2(q)invariant sets of 2-designs with k = 3. We show that this is a complete list. In particular there are no block transitive large sets of t-designs. Moreover, if 1 ~ a < b < c are odd integers such that gcd (a, b) = 1 and ab divides c, then we construct a block transitive Room rectangle based on the 3-subsets of a (7 c + 1)-set whose rows are Steiner triple systems on 7 a points, and whose columns are Steiner triple systems on 7 b points.