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On two non-existence results for Cameron-Liebler k-sets in PG(n,q)

Jan De Beule (UGent) , Jonathan Mannaert and Leo Storme (UGent)
(2025) DESIGNS CODES AND CRYPTOGRAPHY. 93(4). p.1163-1177
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Abstract
This paper focuses on non-existence results for Cameron-Liebler k-sets. A Cameron-Liebler k-set is a collection of k-spaces in PG(n,q) or AG(n,q) admitting a certain parameter x, which is dependent on the size of this collection. One of the main research questions remains the (non-)existence of Cameron-Liebler k-sets with parameter x. This paper improves two non-existence results. First we show that the parameter of a non-trivial Cameron-Liebler k-set in PG(n,q) should be larger than qn-5k/2-1, which is an improvement of an earlier known lower bound. Secondly, we prove a modular equality on the parameter x of Cameron-Liebler k-sets in PG(n,q) with x<q(n-k)-1/q(k+1)-1, n >= 2k+1, n-k+1 >= 7 and n-k even. In the affine case we show a similar result for n-k+1 >= 3 and n-k even. This is a generalization of earlier known modular equalities in the projective and affine case.
Keywords
Cameron-Lieber sets, Boolean functions, Modular equality, LINE CLASSES, MODULAR EQUALITY, SPACES, FAMILY

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MLA
De Beule, Jan, et al. “On Two Non-Existence Results for Cameron-Liebler k-Sets in PG(n,q).” DESIGNS CODES AND CRYPTOGRAPHY, vol. 93, no. 4, 2025, pp. 1163–77, doi:10.1007/s10623-024-01505-8.
APA
De Beule, J., Mannaert, J., & Storme, L. (2025). On two non-existence results for Cameron-Liebler k-sets in PG(n,q). DESIGNS CODES AND CRYPTOGRAPHY, 93(4), 1163–1177. https://doi.org/10.1007/s10623-024-01505-8
Chicago author-date
De Beule, Jan, Jonathan Mannaert, and Leo Storme. 2025. “On Two Non-Existence Results for Cameron-Liebler k-Sets in PG(n,q).” DESIGNS CODES AND CRYPTOGRAPHY 93 (4): 1163–77. https://doi.org/10.1007/s10623-024-01505-8.
Chicago author-date (all authors)
De Beule, Jan, Jonathan Mannaert, and Leo Storme. 2025. “On Two Non-Existence Results for Cameron-Liebler k-Sets in PG(n,q).” DESIGNS CODES AND CRYPTOGRAPHY 93 (4): 1163–1177. doi:10.1007/s10623-024-01505-8.
Vancouver
1.
De Beule J, Mannaert J, Storme L. On two non-existence results for Cameron-Liebler k-sets in PG(n,q). DESIGNS CODES AND CRYPTOGRAPHY. 2025;93(4):1163–77.
IEEE
[1]
J. De Beule, J. Mannaert, and L. Storme, “On two non-existence results for Cameron-Liebler k-sets in PG(n,q),” DESIGNS CODES AND CRYPTOGRAPHY, vol. 93, no. 4, pp. 1163–1177, 2025.
@article{01JYKEV75FD4RVW4T5V7BAE6X1,
  abstract     = {{This paper focuses on non-existence results for Cameron-Liebler k-sets. A Cameron-Liebler k-set is a collection of k-spaces in PG(n,q) or AG(n,q) admitting a certain parameter x, which is dependent on the size of this collection. One of the main research questions remains the (non-)existence of Cameron-Liebler k-sets with parameter x. This paper improves two non-existence results. First we show that the parameter of a non-trivial Cameron-Liebler k-set in PG(n,q) should be larger than qn-5k/2-1, which is an improvement of an earlier known lower bound. Secondly, we prove a modular equality on the parameter x of Cameron-Liebler k-sets in PG(n,q) with x<q(n-k)-1/q(k+1)-1, n >= 2k+1, n-k+1 >= 7 and n-k even. In the affine case we show a similar result for n-k+1 >= 3 and n-k even. This is a generalization of earlier known modular equalities in the projective and affine case.}},
  author       = {{De Beule, Jan and Mannaert, Jonathan and Storme, Leo}},
  issn         = {{0925-1022}},
  journal      = {{DESIGNS CODES AND CRYPTOGRAPHY}},
  keywords     = {{Cameron-Lieber sets,Boolean functions,Modular equality,LINE CLASSES,MODULAR EQUALITY,SPACES,FAMILY}},
  language     = {{eng}},
  number       = {{4}},
  pages        = {{1163--1177}},
  title        = {{On two non-existence results for Cameron-Liebler k-sets in PG(n,q)}},
  url          = {{http://doi.org/10.1007/s10623-024-01505-8}},
  volume       = {{93}},
  year         = {{2025}},
}
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