We determine the minimum possible column multiplicity of even, doubly-, and triply-even codes given their length. This refines a classification result for the possible lengths of $q^r$-divisible codes over $\mathbb{F}_q$. We also give a few computational results for field sizes $q>2$. Non-existence results of divisible codes with restricted column multiplicities for a given length have applications e.g. in Galois geometry and can be used for upper bounds on the maximum cardinality of subspace codes.

翻译：我们确定了给定长度下偶、双偶和三偶码的最小可能列重数。这细化了关于 $\mathbb{F}_q$ 上 $q^r$ 可整除码的可能长度的分类结果。我们还给出了域大小 $q>2$ 的一些计算结论。对于给定长度下具有限制列重数的可整除码的非存在性结果在伽罗瓦几何等领域有应用，并可用于子空间码最大基数的上界。