Open Access
March, 1950 The Impossibility of Certain Symmetrical Balanced Incomplete Block Designs
S. S. Shrikhande
Ann. Math. Statist. 21(1): 106-111 (March, 1950). DOI: 10.1214/aoms/1177729889


An arrangement of $v$ varieties or treatments in $b$ blocks of size $k, (k < v),$ is known as a balanced incomplete block design if every variety occurs in $r$ blocks and any two varieties occur together in $\lambda$ blocks. These parameters obviously satisfy the equations \begin{equation*}\tag{1} bk = vr\end{equation*}\begin{equation*}\tag{2} \lambda(v - 1) = r(k - 1)\end{equation*} Fisher [1] has also proved that the inequality \begin{equation*}\tag{3} b \geq v, \quad r \geq k\end{equation*} must hold. If $v, b, r, k$ and $\lambda$ are positive integers satisfying (1), (2) and (3), then a balanced incomplete block design with these parameters possibly exists, but the actual existence of a combinatorial solution is not ensured. These conditions are thus necessary but not sufficient for the existence of a design. Fisher and Yates in their tables [2] have listed all designs with $r \leq 10$ and given combinatorial solutions, where known. A balanced incomplete block design in which $b = v$, and hence $r = k$ is called a symmetrical balanced incomplete block design. The impossibility of the symmetrical designs with parameters $v = b = 22, r = k = 7, \lambda = 2$ and $v = b = 29, r = k = 8, \lambda = 2$ was first demonstrated by Hussain [3], [4] essentially by the method of enumeration. The object of the present note is to give an alternative simple proof of the impossibility of these designs and to show that the only unknown remaining symmetrical design in Fisher and Yates' tables, viz. $v = b = 46, r = k = 10, \lambda = 2,$ is definitely impossible. Symmetrical designs with $\lambda \leq 5, r, k \leq 20$, which are impossible combinatorially, are also listed.


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S. S. Shrikhande. "The Impossibility of Certain Symmetrical Balanced Incomplete Block Designs." Ann. Math. Statist. 21 (1) 106 - 111, March, 1950.


Published: March, 1950
First available in Project Euclid: 28 April 2007

zbMATH: 0040.36203
MathSciNet: MR32552
Digital Object Identifier: 10.1214/aoms/1177729889

Rights: Copyright © 1950 Institute of Mathematical Statistics

Vol.21 • No. 1 • March, 1950
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