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. For the computational results we partially use extensive integer linear programming computations.

翻译：我们考虑确定射影空间 $\operatorname{PG}(r-1,q)$ 中最多包含 $n_q(r,h,f;s)$ 个 $(h-1)$-子空间的几何问题，使得每个余维数为 $f$ 的子空间至多包含 $s$ 个元素。在编码论术语中，我们处理的是具有大 $f$ 阶广义汉明重量的加性码。同时，我们还考虑对偶问题：确定 $\operatorname{PG}(r-1,q)$ 中最少包含 $b_q(r,h,f;s)$ 个 $(h-1)$-子空间，使得每个余维数为 $f$ 的子空间至少包含 $s$ 个元素。我们完全确定了 $b_2(5,2,2;s)$ 作为 $s$ 的函数表达式。此外，我们还给出了其他参数下的界与构造。在计算结果中，我们部分使用了大规模整数线性规划计算。