Higgledy-piggledy subspaces and uniform subspace designs

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 set $\mathcal{H}$ of higgledy-piggledy $k$-subspaces has to contain more than $\min \mathrm{\mid }\mathbb{F}\mathrm{\mid },{\sum }_{i=0}^k\lfloor \frac{d-k+i}{i+1}\rfloor$ 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 $H_1,\mathrm{.}\mathrm{.},H_N\le {\mathbb{F}}^m$ 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 ${\sum }_{i=1}^N\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(H_i\cap W)\le A$ for $\mathrm{\forall }W\le {\mathbb{F}}^m$ of rank $s$. We prove that if $m=r+s$ then the dual ($\{H_1^{\mathrm{\perp }},\mathrm{.}\mathrm{.}\mathrm{.},H_N^{\mathrm{\perp }}\}$) 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 $A\ge \min \mathrm{\mid }\mathbb{F}\mathrm{\mid },{\sum }_{i=0}^{r-1}\lfloor \frac{s+i}{i+1}\rfloor$ for $r$-uniform weak or strong $(s,A)$ subspace designs in ${\mathbb{F}}^{r+s}$. We show that the $r$-uniform strong $(s,r\cdot s+\left(\genfrac{}{}{0px}{}{r}{2}\right))$ subspace design constructed by Guruswami and Kopprty (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.