Higgledy-piggledy sets in projective spaces of small dimension

This work focuses on higgledy-piggledy sets of $k$-subspaces in $\text{PG}(N,q)$, 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 $\text{PG}(N,q)$ spans $\kappa$ itself. We highlight three methods to construct small higgledy-piggledy sets of $k$-subspaces and discuss, for $k\in\{1,N-2\}$, 'optimal' sets that cover the smallest possible number of points. Furthermore, we investigate small non-trivial higgledy-piggledy sets in $\text{PG}(N,q)$, $N\leqslant5$. Our main result is the existence of six lines of $\text{PG}(4,q)$ in higgledy-piggledy arrangement, two of which intersect. Exploiting the construction methods mentioned above, we also show the existence of six planes of $\text{PG}(4,q)$ in higgledy-piggledy arrangement, two of which maximally intersect, as well as the existence of two higgledy-piggledy sets in $\text{PG}(5,q)$ consisting of eight planes and seven solids, respectively. Finally, we translate these geometrical results to a coding- and graph-theoretical context.Comment: [v1] 21 pages, 1 figure [v2] 21 pages, 1 figure: corrected minor details, updated bibliograph

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 $\min{|\mathbb{F}|,\sum_{i=0}^k\lfloor\frac{d-k+i}{i+1}\rfloor}$ 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 $H_1,..,H_N\le\mathbb{F}^m$ 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 ($\{H_1^\bot,...,H_N^\bot\}$) 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 $A\ge\min{|\mathbb{F}|,\sum_{i=0}^{r-1}\lfloor\frac{s+i}{i+1}\rfloor}$ for $r$-uniform weak or strong $(s,A)$ subspace designs in $\mathbb{F}^{r+s}$. We show that the $r$-uniform strong $(s,r\cdot s+\binom{r}{2})$ 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.Comment: 27 pages. Submitted to Designs Codes and Cryptograph

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 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 H of k-subspaces has to contain more than min {vertical bar F vertical bar, Sigma(k)(i=0) left perpendicular d-k+I/i+1 right perpendicular} elements. We also prove that H has to contain more than (k + 1) . (d -k) elements if the field F is algebraically closed. An r-uniform weak (s, A) subspace design is a set of linear subspaces H-1 ,..., H-N = min { vertical bar F vertical bar, Sigma(r=1)(i=0) left perpendicular s+i/i+1right perpendicular} for r-uniform weak or strong (s, A) subspace designs in Fr+ s. We show that the r-uniform strong (s, r . s + ((r)(2))) subspace design constructed by Guruswami and Kopparty (based on multiplicity codes) has parameter 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) . (d -k) + 1 over algebraically closed field is tight

Resolving sets for higher dimensional projective spaces

Lower and upper bounds on the size of resolving sets for the point-hyperplane incidence graph of the finite projective space PG(n, q) are presented

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 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 H of k-subspaces has to contain more than min {vertical bar F vertical bar, Sigma(k)(i=0) left perpendicular d-k+I/i+1 right perpendicular} elements. We also prove that H has to contain more than (k + 1) . (d -k) elements if the field F is algebraically closed. An r-uniform weak (s, A) subspace design is a set of linear subspaces H-1 ,..., H-N = min { vertical bar F vertical bar, Sigma(r=1)(i=0) left perpendicular s+i/i+1right perpendicular} for r-uniform weak or strong (s, A) subspace designs in Fr+ s. We show that the r-uniform strong (s, r . s + ((r)(2))) subspace design constructed by Guruswami and Kopparty (based on multiplicity codes) has parameter 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) . (d -k) + 1 over algebraically closed field is tight

Small Strong Blocking Sets by Concatenation

Strong blocking sets and their counterparts, minimal codes, attracted lots of attention in the last years. Combining the concatenating construction of codes with a geometric insight into the minimality condition, we explicitly provide infinite families of small strong blocking sets, whose size is linear in the dimension of the ambient projective spaces. As a byproduct, small saturating sets are obtained.Comment: 16 page