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On the existence of simple BIBDs with number of elements a prime power. (English) Zbl 1258.05009

Summary: We show that, when the number of elements is a prime power \(q\), in many situations the necessary conditions
\[ \begin{aligned} \lambda(q-1) &\equiv 0 \mod (k-1), \tag{1}\\ \lambda q(q-1) &\equiv 0 \mod k(k-1), \text{ and} \tag{2}\\ z\lambda &\leq \binom{q-2}{k-2} \tag{3} \end{aligned} \]
are also sufficient for the existence of a \((q,k,\lambda)\) simple balanced incomplete block design.

MSC:

05B05 Combinatorial aspects of block designs
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References:

[1] Abel, The CRC Handbook of Combinatorial Designs pp 392– (2007)
[2] Beth, Design Theory (1999)
[3] Colbourn, The CRC Handbook of Combinatorial Designs (2007) · Zbl 1101.05001
[4] Sun, From planar nearrings to generating blocks, Taiwanese J Math 14 (5) pp 1713– (2010) · Zbl 1228.05081
[5] Wilson, Cyclotomy and difference families in elementary abelian groups, J Number Theory 4 pp 17– (1972) · Zbl 0259.05011
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