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Modified group divisible designs. (English) Zbl 0702.05014

A design of the form (X,C,B) is considered where X is a finite set of points, C is a parallel class of subsets of X called groups and B is a family of subsets of X called blocks. Let m, k, \(\lambda\) and v be positive integers. A design (X,C,B) is called a modified group divisible design and denoted by MGD[k,\(\lambda\),m,v] if (i) \(| X| =v\); (ii) \(| G_ i| =m\) for every \(G_ i\) in C; (iii) \(| B_ j| =k\) for every \(B_ j\) in B; (iv) \(| G_ i\cap G_ j| \leq 1\) for every \(G_ i\) in C and every \(B_ j\) in B; (v) every \(\{\) x,y\(\}\subset X\) such that x and y are neither in the same group nor in the same row is contained in exactly \(\lambda\) blocks of B. The present paper provides the following main theorem: Let m, \(\lambda\) and v be positive integers. The necessary and sufficient conditions for the existence of a modified group divisible design MGD[3,\(\lambda\),m,v] are that \(v\equiv 0\) (mod m), \(v\geq 3m\), \(m\geq 3\), \(\lambda (v+1-m-n)\equiv 0\) (mod 2) and \(\lambda v(v+1-m-n)\equiv 0\) (mod 6).
Reviewer: S.Kageyama

MSC:

05B05 Combinatorial aspects of block designs
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