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Group divisible designs of four groups and block size five with configuration \((1, 1, 1, 2)\). (English) Zbl 1440.05046

Summary: We present constructions and results about GDDs with four groups and block size five in which each block has configuration \((1, 1, 1, 2)\), that is, each block has exactly one point from three of the four groups and two points from the fourth group. We provide the necessary conditions of the existence of a GDD \((n, 4, 5; \lambda_1, \lambda_2)\) with configuration \((1, 1, 1, 2)\), and show that the necessary conditions are sufficient for a GDD \((n, 4, 5; \lambda_1, \lambda_2)\) with configuration \((1, 1, 1, 2)\) if \(n \not\equiv 0 \pmod 6\), respectively. We also show that a GDD \((n, 4, 5; 2n, 6(n - 1))\) with configuration \((1, 1, 1, 2)\) exists, and provide constructions for a GDD \((n = 2t, 4, 5; n, 3(n - 1))\) with configuration \((1, 1, 1, 2)\) where \(n \neq 12\), and a GDD \((n = 6t, 4, 5; 4t, 2(6t - 1))\) with configuration \((1, 1, 1, 2)\) where \(n \neq 6\) and 18, respectively.

MSC:

05B05 Combinatorial aspects of block designs
05B30 Other designs, configurations
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References:

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