Sets of subspaces with restricted hyperplane intersection numbers

Abstract

Let $\mathcal{X}$ be a set of $(h-1)$-dimensional subspaces of $\mathrm{PG}(kh-1,q)$ with the property that every hyperplane contains at most $t$ elements of $\mathcal{X}$. We prove the upper bound $|\mathcal{X}| \leq (t-k+2)q^h + t$, and characterise the structure of $\mathcal{X}$ in the case of equality. We call sets attaining this bound \emph{length-maximal}. For $k=3$, such sets are known as maximal arcs and have been well-studied. They are known to exist for $t<q^h$ if and only if $q$ is even and $t$ divides $q^h$. For $k=4$ and $q>2$, we show that any length-maximal set must satisfy $t = q^h+1$ and that every hyperplane is either a $t$-secant or a $1$-secant. For $k \geq 5$ and $q>2$, no length-maximal set exists. In the language of additive codes, these results assert that additive two-weight codes over $\mathbb{F}_{q^h}$ attaining the natural Griesmer-type bound do not exist when the code dimension is $5$ or more and $q>2$.