# American Institute of Mathematical Sciences

August  2019, 13(3): 477-503. doi: 10.3934/amc.2019030

## A spectral characterisation of $t$-designs and its applications

 1 Department of Mathematics, Pusan National University, Republic of Korea 2 Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China 3 Korea Institute for Advanced Study (KIAS), Seoul, Republic of Korea

* Corresponding author

Received  August 2018 Published  April 2019

Fund Project: C. Ding was supported by The Hong Kong Grants Council, Proj. No. 16300418. J. Y. Hyun was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (NRF-2017R1D1A1B05030707)

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.

Citation: Eun-Kyung Cho, Cunsheng Ding, Jong Yoon Hyun. A spectral characterisation of $t$-designs and its applications. Advances in Mathematics of Communications, 2019, 13 (3) : 477-503. doi: 10.3934/amc.2019030
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##### References:
Spectrum of $f_{ {\mathbb{D}}}$
 Weight of $w$ Multiset $\{ \hat{f}_{ {\mathbb{D}}}(w) \}$ $0, 12$ $\{ 132 \}$ $1, 11$ $\{ 0^{12} \}$ $2, 10$ $\{ -12^{66} \}$ $3, 9$ $\{ 0^{220} \}$ $4, 8$ $\{ 4^{495} \}$ $5, 7$ $\{ 0^{792} \}$ $6$ $\{ -12^{792}, 52^{132}\}$
 Weight of $w$ Multiset $\{ \hat{f}_{ {\mathbb{D}}}(w) \}$ $0, 12$ $\{ 132 \}$ $1, 11$ $\{ 0^{12} \}$ $2, 10$ $\{ -12^{66} \}$ $3, 9$ $\{ 0^{220} \}$ $4, 8$ $\{ 4^{495} \}$ $5, 7$ $\{ 0^{792} \}$ $6$ $\{ -12^{792}, 52^{132}\}$
Weight distribution
 Weight $w$ No. of codewords $A_w$ $0$ $1$ $132$ $1$ $2^{11}-12$ $924$ $2^{11}$ $6143$ $2^{11}+4$ $990$ $2^{11}+52$ $132$ $2^{11}+132$ $1$
 Weight $w$ No. of codewords $A_w$ $0$ $1$ $132$ $1$ $2^{11}-12$ $924$ $2^{11}$ $6143$ $2^{11}+4$ $990$ $2^{11}+52$ $132$ $2^{11}+132$ $1$
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