We consider the geometric problem of determining the maximum number $n_q(r,h,f;s)$ of $(h-1)$-spaces in the projective space $\operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ does contain at most $s$ elements. In coding theory terms we are dealing with additive codes that have a large $f$th generalized Hamming weight. We also consider the dual problem of the minimum number $b_q(r,h,f;s)$ of $(h-1)$-spaces in $\operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ contains at least $s$ elements. We fully determine $b_2(5,2,2;s)$ as a function of $s$. We additionally give bounds and constructions for other parameters.

翻译：我们考虑一个几何问题：确定射影空间 $\operatorname{PG}(r-1,q)$ 中 $(h-1)$-维子空间的最大数量 $n_q(r,h,f;s)$，使得每个余维数为 $f$ 的子空间至多包含 $s$ 个元素。从编码理论的角度，这对应于具有较大第 $f$ 广义汉明重量的加性码。我们还考虑了其对偶问题：确定 $\operatorname{PG}(r-1,q)$ 中 $(h-1)$-维子空间的最小数量 $b_q(r,h,f;s)$，使得每个余维数为 $f$ 的子空间至少包含 $s$ 个元素。我们完整确定了 $b_2(5,2,2;s)$ 作为 $s$ 的函数。此外，我们还针对其他参数给出了界和构造。