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July, 1975 Duals of Balanced Incomplete Block Designs Derived from an Affine Geometry
Noboru Hamada, Fumikazu Tamari
Ann. Statist. 3(4): 926-938 (July, 1975). DOI: 10.1214/aos/1176343193


It is well known at by identifying the points of an affine geometry AG (t, q) with treatments and identifying the $\mu$-flats $(1\leqq\mu<t)$ of AG (t, q) with blocks, a BIB design denoted by AG (t, q): $\mu$ is derived from AG (t, q) where q is a prime or a prime power. In this paper, we introduce a new association scheme called an affine geometrical association scheme and show that the dual of the BIB design AG (t, q): $\mu$ is an affine geometrical type PBIB design with $m=min{2\mu+1,2(t-\mu)}$ associate classes. It is also shown that in the case $\mu=1$ and $t\geqq3$, the number of the associate classes of this dual design can be reduced from three to two but it is not reducible except for the above case. From those results, we can get a new series of PBIB designs.


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Noboru Hamada. Fumikazu Tamari. "Duals of Balanced Incomplete Block Designs Derived from an Affine Geometry." Ann. Statist. 3 (4) 926 - 938, July, 1975.


Published: July, 1975
First available in Project Euclid: 12 April 2007

zbMATH: 0309.62058
MathSciNet: MR375672
Digital Object Identifier: 10.1214/aos/1176343193

Primary: 62K10 (05B05)
Secondary: 05B25

Keywords: Balanced incomplete block design , dual design , finite geometries , partially balanced incomplete block design , reduction of associate classes

Rights: Copyright © 1975 Institute of Mathematical Statistics

Vol.3 • No. 4 • July, 1975
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