A spectral characterisation of t-designs

Cunsheng Ding

Published 2017-06-01Version 1

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. These approaches are not effective as no infinite family of $t$-designs is constructed for $t \geq 5$ with them. 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. We will determine the spectra of the characteristic function of $(n-2)/2$-$(n, n/2, 1)$ Steiner systems and prove properties of such designs.