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Constructing \(t\)-designs from \(t\)-wise balanced designs. (English) Zbl 1161.05010

Let \(X\) be a finite set of nonzero cardinality \(v\) and \(B\) be a nonempty collection of subsets of \(X\). If each subset of \(X\) with cardinality \(t>0\) is contained in exactly \(\lambda>0\) of the elements of \(B\), then this structure is called a \(t\)-wise balanced design. This paper provides a procedure for constructing from such a design, a balanced design (called a \(t\)-design) with all the elements of \(B\) having the same cardinality. It is shown that the automorphism group of the new design contains as a subgroup, the automorphism group of the original design. The construction is also used to produce a \(t\)-design for a smaller set (\(X\) with one point removed). The authors state that their construction was found as a result of looking for 2-designs with repeated blocks to help fill up the catalogue by D. A. Preece [A selection of BIBDs with repeated blocks, \(r \leq 20\), \(\text{gcd}(b,r,\lambda)=1\), 2003 (Preprint)]. They also provide examples constructed and tested with the DESIGN package in GAP (http://www.gap-system.org, 2004).

MSC:

05B05 Combinatorial aspects of block designs

Software:

GAP; DESIGN
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Full Text: DOI

References:

[1] (Colbourn, C. J.; Dinitz, J. H., The CRC Handbook of Combinatorial Designs (1996), CRC Press) · Zbl 0836.00010
[2] GAP — Groups, Algorithms, and Programming, Version 4.4, Aachen, St Andrews (2004)
[3] Hughes, D. R.; Piper, F. C., Design Theory (1988), Cambridge University Press · Zbl 0659.05008
[4] D.A. Preece, A selection of BIBDs with repeated blocks, \(r \leq 20\gcd ( b , r , \lambda ) = 1\); D.A. Preece, A selection of BIBDs with repeated blocks, \(r \leq 20\gcd ( b , r , \lambda ) = 1\)
[5] Soicher, L. H., The DESIGN package for GAP, Version 1.1 (2004)
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