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arXiv · 1706.00180

A spectral characterisation of t-designs and its applications

Eun-Kyung Cho·Cunsheng Ding·Jong Yoon Hyun
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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.

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Eun-Kyung Cho, Cunsheng Ding, Jong Yoon Hyun. 2018-06-09. A spectral characterisation of t-designs and its applications. https://arxiv.org/abs/1706.00180

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