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 spanning sets attaining this bound \emph{length-maximal}. For $k=3$, these sets are higher-dimensional analogues of maximal arcs; when $h=1$ they are precisely the maximal arcs of $\mathrm{PG}(2,q)$, which for $t 2$, 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. Such sets exist for all $q$ and $h$, arising as field reductions of ovoids of $\mathrm{PG}(3,q^h)$. For $k \geq 5$ and $q^h>3$, no length-maximal set exists. The case $q^h=3$, that is $(q,h)=(3,1)$, is exceptional: there, examples arising from the ternary Golay code exist for $k=5$ and $k=6$, while none exist for $k\ge7$. In the language of additive codes, these results assert that additive codes over $\mathbb{F}_{q^h}$ attaining the natural Griesmer-type bound do not exist when the code dimension is $5$ or more and $q^h>3$, apart from these two sporadic $\mathbb{F}_3$ examples. We also determine the strongly regular graphs associated with length-maximal sets.