  There are two standard approaches to the construction of $t$-designs. The
first one is based on permutation group actions on certain base blocks. The
second one is based on coding theory. The objective of this paper is to give a
spectral characterisation of all $t$-designs by introducing a characteristic
Boolean function of a $t$-design. The spectra of the characteristic functions
of $(n-2)/2$-$(n, n/2, 1)$ Steiner systems are determined and properties of
such designs are proved. Delsarte's characterisations of orthogonal arrays and
$t$-designs, which are two special cases of Delsarte's characterisation of
$T$-designs in association schemes, are slightly extended into two spectral
characterisations. Another characterisation of $t$-designs by Delsarte and
Seidel is also extended into a spectral one. These spectral characterisations
are then compared with the new spectral characterisation of this paper.
