The Impossibility of Certain Symmetrical Balanced Incomplete Block Designs

Abstract

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.