Symmetric balanced incomplete block design ($b=t,r=k$)

A balanced incomplete block design is symmetric when its number of blocks equals its number of treatments. The identity $bk=tr$ then gives $r=k$. Its square incidence matrix of a set system $N$ obeys $N{N}^T=(r-\lambda )I+\lambda J$, with eigenvalues $r-\lambda$ of multiplicity $t-1$ and $r^2$ of multiplicity one.

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.