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 $$\mathsf {PG}(d,\mathbb {F})$$PG(d,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 $$\mathcal {H}$$H of k-subspaces has to contain more than$$\min \left\{ |\mathbb {F}|,\sum _{i=0}^k\left\lfloor \frac{d-k+i}{i+1}\right\rfloor \right\} $$min|F|, i=0kd-k+ii+1 elements. We also prove that $$\mathcal {H}$$H has to contain more than$$(k+1)\cdot (d-k)$$(k+1)·(d-k) elements if the field $$\mathbb {F}$$F is algebraically closed. An r-uniform weak (s, A) subspace design is a set of linear subspaces $$H_1,\ldots,H_N\le \mathbb {F}^m$$H1,HN≤Fm each of rank r such that each linear subspace $$W\le \mathbb {F}^m$$W≤Fm of rank s meets at most A among them. This subspace design is an r-uniform strong (s, A) subspace design if $$\sum _{i=1}^N\mathrm {rank}(H_i\cap W)\le A$$ i=1Nrank(Hi W)≤A for $$\forall W\le \mathbb {F}^m$$ W≤Fm of rank s. We prove that if $$m=r+s$$m=r+s then the dual ($$\{H_1^\bot,\dots,H_N^\bot \}$${H1,HN }) 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 $$A\ge \min \left\{ |\mathbb {F}|,\sum _{i=0}^{r-1}\left\lfloor \frac{s+i}{i+1}\right\rfloor \right\} $$A min|F|, i=0r-1s+ii+1 for r-uniform weak or strong (s, A) subspace designs in $$\mathbb {F}^{r+s}$$Fr+s. We show that the r-uniform strong $$(s,r\cdot s+\left( {\begin{array}{c}r\\ 2\end{array}}\right) )$$(s,r·s+r2) subspace design constructed by Guruswami and Kopparty (based on multiplicity codes) has parameter $$A=r\cdot s$$A=r·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 $$(k+1)\cdot (d-k)+1$$(k+1)·(d-k)+1 over algebraically closed field is tight.