# A spectral characterisation of t-designs and its applications

## TL;DR

This paper introduces a spectral characterization method for all t-designs using Boolean functions, extending existing algebraic and combinatorial frameworks, and applies it to Steiner systems and orthogonal arrays.

## Contribution

It provides a new spectral characterization of t-designs via Boolean functions, extending Delsarte's and Seidel's characterizations, and applies it to Steiner systems and orthogonal arrays.

## Key findings

- Spectra of characteristic functions for Steiner systems are determined.
- Spectral characterizations extend existing algebraic frameworks.
- New insights into properties of t-designs are proved.

## Abstract

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.