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Optimal and extremal graphical designs on regular graphs associated with classical parameters

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Abstract

Graphical designs are the extension of spherical designs to finite graphs from the viewpoint of quadrature formulas. In this paper, we investigate optimal graphical designs on hypercubes, especially the conjecture proposed by Babecki that the Hamming code is an optimal graphical design on the hypercube. We prove that this conjecture is not true using certain binary t-error-correcting BCH codes. We also obtain extremal graphical designs on the furthest distance graph of 13 families of distance-regular graphs with classical parameters. This generalizes the result that any 1-intersecting family achieving Erdös–Ko–Rado type bound is an extremal graphical design on the Kneser graph.

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Acknowledgements

The author would like to thank Eiichi Bannai and Yaokun Wu for many valuable comments and discussions. The research is supported by National Natural Science Foundation of China No. 11801353.

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Correspondence to Yan Zhu.

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Communicated by J. H. Koolen.

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Zhu, Y. Optimal and extremal graphical designs on regular graphs associated with classical parameters. Des. Codes Cryptogr. 91, 2737–2754 (2023). https://doi.org/10.1007/s10623-023-01231-7

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