Higgledy-piggledy subspaces and uniform subspace designs

Abstract

In this article, we investigate collections of ‘well-spread-out’ projective (and linear) subspaces. Projective k-subspaces in $\mathsf {PG}(d,\mathbb {F})$ are in ‘higgledy-piggledy arrangement’ if they meet each projective subspace of co-dimension k in a generator set of points. We prove that the higgledy-piggledy set $\mathcal {H}$ of k-subspaces has to contain more than $\min \left\{ |\mathbb {F}|,\sum _{i=0}^k\left\lfloor \frac{d-k+i}{i+1}\right\rfloor \right\}$ elements. We also prove that $\mathcal {H}$ has to contain more than $(k+1)\cdot (d-k)$ elements if the field $\mathbb {F}$ is algebraically closed. An r-uniform weak (s, A) subspace design is a set of linear subspaces $H_1,\ldots ,H_N\le \mathbb {F}^m$ each of rank r such that each linear subspace $W\le \mathbb {F}^m$ of rank s meets at most A among them. This subspace design is an r-uniform strong (s, A) subspace design if $\sum _{i=1}^N\mathrm {rank}(H_i\cap W)\le A$ for $\forall W\le \mathbb {F}^m$ of rank s. We prove that if $m=r+s$ then the dual ($\{H_1^\bot ,\dots ,H_N^\bot \}$) of an r-uniform weak (strong) subspace design of parameter (s, A) is an s-uniform weak (strong) subspace design of parameter (r, A). We show the connection between uniform weak subspace designs and higgledy-piggledy subspaces proving that $A\ge \min \left\{ |\mathbb {F}|,\sum _{i=0}^{r-1}\left\lfloor \frac{s+i}{i+1}\right\rfloor \right\}$ for r-uniform weak or strong (s, A) subspace designs in $\mathbb {F}^{r+s}$. We show that the r-uniform strong $(s,r\cdot s+\left( {\begin{array}{c}r\\ 2\end{array}}\right) )$ subspace design constructed by Guruswami and Kopparty (based on multiplicity codes) has parameter $A=r\cdot s$ if we consider it as a weak subspace design. We give some similar constructions of weak and strong subspace designs (and higgledy-piggledy subspaces) and prove that the lower bound $(k+1)\cdot (d-k)+1$ over algebraically closed field is tight.