Lines in Higgledy-Piggledy Arrangement

  • Szabolcs L. Fancsali
  • Péter Sziklai
Keywords: well-spread-out (`higgledy-piggledy') lines, subspace designs, Grassmann variety, Plücker co-ordinates


In this article, we examine sets of lines in $\mathsf{PG}(d,\mathbb{F})$ meeting each hyperplane in a generator set of points. We prove that such a set has to contain at least $\lfloor1.5d\rfloor$ lines if the field $\mathbb{F}$ has at least $\lfloor1.5d\rfloor$ elements, and at least $2d-1$ lines if the field $\mathbb{F}$ is algebraically closed. We show that suitable $2d-1$ lines constitute such a set (if $|\mathbb{F}|\ge2d-1$), proving that the lower bound is tight over algebraically closed fields. At last, we will see that the strong $(s,A)$ subspace designs constructed by Guruswami and Kopparty have better (smaller) parameter $A$ than one would think at first sight.

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