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Degree 2 Boolean functions on Grassmann graphs

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Abstract
We investigate the existence of Boolean degree d functions on the Grassmann graph of k-spaces in the vector space F_{q^n}. For d=1 several non-existence and classification results are known, and no non-trivial examples are known for n ≥ 5. This paper focusses on providing a list of examples on the case d=2 in general dimension and in particular for (n, k)=(6,3) and (n,k) = (8, 4). We also discuss connections to the analysis of Boolean functions, regular sets/equitable bipartitions/perfect 2-colorings in graphs, q-analogs of designs, and permutation groups. In particular, this represents a natural generalization of Cameron-Liebler line classes.
Keywords
Computational Theory and Mathematics, Geometry and Topology, Theoretical Computer Science, Applied Mathematics, Discrete Mathematics and Combinatorics, DESIGNS, ORBITS, SPACES, SETS

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MLA
De Beule, Jan, et al. “Degree 2 Boolean Functions on Grassmann Graphs.” ELECTRONIC JOURNAL OF COMBINATORICS, vol. 30, no. 1, 2023, doi:10.37236/11040.
APA
De Beule, J., D’haeseleer, J., Ihringer, F., & Mannaert, J. (2023). Degree 2 Boolean functions on Grassmann graphs. ELECTRONIC JOURNAL OF COMBINATORICS, 30(1). https://doi.org/10.37236/11040
Chicago author-date
De Beule, Jan, Jozefien D’haeseleer, Ferdinand Ihringer, and Jonathan Mannaert. 2023. “Degree 2 Boolean Functions on Grassmann Graphs.” ELECTRONIC JOURNAL OF COMBINATORICS 30 (1). https://doi.org/10.37236/11040.
Chicago author-date (all authors)
De Beule, Jan, Jozefien D’haeseleer, Ferdinand Ihringer, and Jonathan Mannaert. 2023. “Degree 2 Boolean Functions on Grassmann Graphs.” ELECTRONIC JOURNAL OF COMBINATORICS 30 (1). doi:10.37236/11040.
Vancouver
1.
De Beule J, D’haeseleer J, Ihringer F, Mannaert J. Degree 2 Boolean functions on Grassmann graphs. ELECTRONIC JOURNAL OF COMBINATORICS. 2023;30(1).
IEEE
[1]
J. De Beule, J. D’haeseleer, F. Ihringer, and J. Mannaert, “Degree 2 Boolean functions on Grassmann graphs,” ELECTRONIC JOURNAL OF COMBINATORICS, vol. 30, no. 1, 2023.
@article{01H25EGAT6H428A41Q55GWBA8E,
  abstract     = {{We investigate the existence of Boolean degree d functions on the Grassmann graph of k-spaces in the vector space F_{q^n}. For d=1 several non-existence and classification results are known, and no non-trivial examples are known for n ≥ 5. This paper focusses on providing a list of examples on the case d=2 in general dimension and in particular for (n, k)=(6,3) and (n,k) = (8, 4). We also discuss connections to the analysis of Boolean functions, regular sets/equitable bipartitions/perfect 2-colorings in graphs, q-analogs of designs, and permutation groups. In particular, this represents a natural generalization of Cameron-Liebler line classes.}},
  articleno    = {{P1.31}},
  author       = {{De Beule, Jan and D'haeseleer, Jozefien and Ihringer, Ferdinand and Mannaert, Jonathan}},
  issn         = {{1077-8926}},
  journal      = {{ELECTRONIC JOURNAL OF COMBINATORICS}},
  keywords     = {{Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics,DESIGNS,ORBITS,SPACES,SETS}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{23}},
  title        = {{Degree 2 Boolean functions on Grassmann graphs}},
  url          = {{http://doi.org/10.37236/11040}},
  volume       = {{30}},
  year         = {{2023}},
}

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