Abstract
Cameron–Liebler sets of subspaces in projective spaces were studied recently by Blokhuis et al. (Des Codes Cryptogr 87:1839–1856, 2019). In this paper, we discuss Cameron–Liebler sets in bilinear forms graphs, obtain several equivalent definitions and present some classification results.
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Notes
For each \((x_1,\ldots ,x_{n})^t\in \mathbb {F}_q^{n}\), let \(V_{(x_1,\ldots ,x_{n})}=\langle e_1+x_1e_{n+1},\ldots ,e_n+x_ne_{n+1},e_{n+2},\ldots ,e_{n+\ell }\rangle .\) Then \(\{V_{(x_1,\ldots ,x_{n})}: (x_1,\ldots ,x_{n})^t\in \mathbb {F}_q^{n}\}\subseteq {\mathcal {M}}(n+\ell -1,\ell -1;n+\ell ,E)\) and \({\mathcal {M}}_n(V_{(x_1,\ldots ,x_{n})})\cap {\mathcal {M}}_n(V_{(y_1,\ldots ,y_{n})})=\emptyset \) for all \((x_1,\ldots ,x_{n})\not =(y_1,\ldots ,y_{n})\).
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Acknowledgements
The author is indebted to the anonymous reviewers for their detailed reports and constructive suggestions. The author thanks Professor Ferdinand Ihringer and Professor Alexander L. Gavrilyuk for their various remarks and suggestions while writing this article. This research is supported by National Natural Science Foundation of China (Grant No. 11971146).
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Communicated by J. D. Key.
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Guo, J. Cameron–Liebler sets in bilinear forms graphs. Des. Codes Cryptogr. 89, 1159–1180 (2021). https://doi.org/10.1007/s10623-021-00864-w
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DOI: https://doi.org/10.1007/s10623-021-00864-w