= Even-order symmetric design square obstruction
{title2=$t\text{ even}\ \Longrightarrow\ r-\lambda\text{ is a square}$}
In a nontrivial <symmetric balanced incomplete block design>, $\det(N)^2=r^2(r-\lambda)^{t-1}$. If $t$ is even, $t-1$ is odd. For every prime, the exponent in the left side is even and that in $r^2$ is even, so its exponent in $r-\lambda$ must be even. Thus $r-\lambda$ is a perfect square. This supplies an elementary necessary existence condition without claiming it is sufficient.
Back to article page
