Block intersection polynomials

Abstract

We introduce the block intersection polynomial, which is constructed using certain information about a block design with respect to a subset S of its point-set, and then provides further information about the number of blocks intersecting S in exactly i points, for i = 0, . . . , |S|. We also discuss some applications of block intersection polynomials, including bounding the multiplicity of a block in a t-(v, k, λ) design and in a resolvable t-(v, k, λ) design.

More on block intersection polynomials and new applications to graphs and block designs

“…We remark that this result is proved for the case when D is a t-design and S is a block in [13], in general for t-designs in [15], and for the case r = 1 for general block designs in [5] (from which we have adapted our proof of Theorem 2.1).…”

“…,n s ] of non-negative integers. Problems of this form arise in the study of block designs, especially the study of t-designs (see [13,8,15,1,2,5]). In [5], Cameron and the present author introduce the block intersection polynomial, and show how this polynomial gives useful information about the solutions to this problem when t is even and specified non-negative integers m 0 , .…”

Designs, Groups and Computing

“…Each vertex of this graph represents a possible H-orbit of blocks, each with the same specified multiplicity, with two distinct vertices not joined by an edge only when the totality of the blocks they represent cannot be a submultiset of the blocks of a required design. Such a non-edge may be a result of user-specified properties of the required designs, or may be determined by applying block intersection polynomials [12,33]. The graph problem is then handled by the GRAPE [35] function CompleteSubgraphsOfGivenSize, which uses a complicated backtrack search.…”

“…This calculation takes about 220 seconds of CPUtime on a 3.1 GHz PC running Linux. [ 4,5 ], [ 4,5 ], [ 4,5 ], [ 4,5 ] ] gap> List(designs,BlockNumbers); [ [ 11 ], [ 10,5 ], [ 10,5 ], [ 10,5 ], [ 10,5 ] ] gap> List(designs,d->Size(AutomorphismGroup(d))); [ 660,6,8,12,120 ] More generally, BlockDesigns can construct subdesigns of a given block design, such that the subdesigns each have the same user-specified properties. Here, a subdesign of ∆ means a block design with the same point set as ∆ and whose block multiset is a submultiset of the blocks of ∆ .…”

“…The total time taken for this calculation is about seven minutes on a 3.1 GHz PC running Linux. [ 4,24 ], [ 8,33 ], [ 12,4 ], [ 16,14 ], [ 24,11 ], [ 32,2 ] A further calculation, taking just one second, shows that no design in L is resolvable; equivalently, no semi-Latin square whose dual is in L is the superposition of Latin squares. …”