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Higgledy-piggledy sets in projective spaces of small dimension

Lins Denaux (UGent)
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Abstract
This work focuses on higgledy-piggledy sets of k-subspaces in PG(N, q), i.e. sets of projective subspaces that are 'well-spread-out'. More precisely, the set of intersection points of these k-subspaces with any (N - k)-subspace kappa of PG(N, q) spans kappa itself. We highlight three methods to construct small higgledy-piggledy sets of k-subspaces and discuss, for k is an element of {1, N - 2}, 'optimal' sets that cover the smallest possible number of points. Furthermore, we investigate small non-trivial higgledy-piggledy sets in PG(N, q), N <= 5. Our main result is the existence of six lines of PG(4, q) in higgledy-piggledy arrangement, two of which intersect. Exploiting the construction methods mentioned above, we also show the existence of six planes of PG(4, q) in higgledy-piggledy arrangement, two of which maximally intersect, as well as the existence of two higgledy-piggledy sets in PG(5, q) consisting of eight planes and seven solids, respectively. Finally, we translate these geometrical results to a coding-and graph-theoretical context.
Keywords
Computational Theory and Mathematics, Geometry and Topology, Theoretical Computer Science, Applied Mathematics, Discrete Mathematics and Combinatorics, SATURATING SETS, EXTERNAL LINES, BLOCKING SETS

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MLA
Denaux, Lins. “Higgledy-Piggledy Sets in Projective Spaces of Small Dimension.” ELECTRONIC JOURNAL OF COMBINATORICS, vol. 29, no. 3, 2022, doi:10.37236/10736.
APA
Denaux, L. (2022). Higgledy-piggledy sets in projective spaces of small dimension. ELECTRONIC JOURNAL OF COMBINATORICS, 29(3). https://doi.org/10.37236/10736
Chicago author-date
Denaux, Lins. 2022. “Higgledy-Piggledy Sets in Projective Spaces of Small Dimension.” ELECTRONIC JOURNAL OF COMBINATORICS 29 (3). https://doi.org/10.37236/10736.
Chicago author-date (all authors)
Denaux, Lins. 2022. “Higgledy-Piggledy Sets in Projective Spaces of Small Dimension.” ELECTRONIC JOURNAL OF COMBINATORICS 29 (3). doi:10.37236/10736.
Vancouver
1.
Denaux L. Higgledy-piggledy sets in projective spaces of small dimension. ELECTRONIC JOURNAL OF COMBINATORICS. 2022;29(3).
IEEE
[1]
L. Denaux, “Higgledy-piggledy sets in projective spaces of small dimension,” ELECTRONIC JOURNAL OF COMBINATORICS, vol. 29, no. 3, 2022.
@article{8762546,
  abstract     = {{This work focuses on higgledy-piggledy sets of k-subspaces in PG(N, q), i.e. sets of projective subspaces that are 'well-spread-out'. More precisely, the set of intersection points of these k-subspaces with any (N - k)-subspace kappa of PG(N, q) spans kappa itself.

We highlight three methods to construct small higgledy-piggledy sets of k-subspaces and discuss, for k is an element of {1, N - 2}, 'optimal' sets that cover the smallest possible number of points.

Furthermore, we investigate small non-trivial higgledy-piggledy sets in PG(N, q), N <= 5. Our main result is the existence of six lines of PG(4, q) in higgledy-piggledy arrangement, two of which intersect. Exploiting the construction methods mentioned above, we also show the existence of six planes of PG(4, q) in higgledy-piggledy arrangement, two of which maximally intersect, as well as the existence of two higgledy-piggledy sets in PG(5, q) consisting of eight planes and seven solids, respectively.

Finally, we translate these geometrical results to a coding-and graph-theoretical context.}},
  articleno    = {{P3.29}},
  author       = {{Denaux, Lins}},
  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,SATURATING SETS,EXTERNAL LINES,BLOCKING SETS}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{25}},
  title        = {{Higgledy-piggledy sets in projective spaces of small dimension}},
  url          = {{http://dx.doi.org/10.37236/10736}},
  volume       = {{29}},
  year         = {{2022}},
}

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