Search

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 Cryptography

In this article we prove a theorem about the number of directions determined by less then $q$ affine points, similar to the result of Blokhuis et al. (in J. Comb. Theory Ser. A 86(1), 187-196, 1999) on the number of directions determined by $q$ affine points., Comment: 12 pages. NOTICE: this is the authors' version of a work that was submitted for publication in J. Algebr. Combin. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms are not reflected in this document. The final publication is available at Springer via doi:10.1007/s10801-012-0357-1

We examine sets of lines in PG(d,F) meeting each hyperplane in a generator set of points. We prove that such a set has to contain at least 1.5d lines if the field F has more than 1.5d elements, and at least 2d-1 lines if the field F is algebraically closed. We show that suitable 2d-1 lines constitute such a set (if |F| > or = 2d-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., Comment: 17 pages