On the structure of 1-designs with at most two block intersection numbers
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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.
KeywordsBlock intersection numbers SOMAs Unrefinable decompositions Ud-types Strongly resolvable designs
AMS Classifications05B05 05B25
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