On the structure of 1-designs with at most two block intersection numbers

We introduce the notion of an unrefinable decomposition of a 1-design with at most two block intersection numbers, which is a certain decomposition of the 1-designs collection of blocks into other 1-designs. We discover an infinite family of 1-designs with at most two block intersection numbers that each have a unique unrefinable decomposition, and we give a polynomial-time algorithm to compute an unrefinable decomposition for each such design from the family. Combinatorial designs from this family include: finite projective planes of order n; SOMAs, and more generally, partial linear spaces of order (s, t) on (s + 1)2 points; as well as affine designs, and more generally, strongly resolvable designs with no repeated blocks. AMS classification. 05B05 (primary), 05B25 (secondary) Keywords. block intersection numbers, SOMAs, unrefinable decompositions, udtypes, strongly resolvable designs. 1 

• Type: Conference
• Year: 2007
• Where: DCC
• Authors: John Arhin
• Keywords: Block Intersection Numbers, Computer Graphics, DCC 2007, Resolvable Designs, Unrefinable Decomposition