The hull of a linear code $C$ is the intersection of $C$ with its dual code. We present and analyze the number of linear $q$-ary codes of the same length and dimension but with different dimensions for their hulls. We prove that for given dimension $k$ and length $n\ge 2k$ the number of all $[n,k]_q$ linear codes with hull dimension $l$ decreases as $l$ increases. We also present classification results for binary and ternary linear codes with trivial hulls (LCD and self-orthogonal) for some values of the length $n$ and dimension $k$, comparing the obtained numbers with the number of all linear codes for the given $n$ and $k$.

翻译：线性码 $C$ 的壳是其与对偶码的交集。我们提出并分析了具有相同长度和维数但壳维数不同的 $q$ 元线性码的数量。我们证明，对于给定维数 $k$ 和长度 $n\ge 2k$，所有壳维数为 $l$ 的 $[n,k]_q$ 线性码的数量随着 $l$ 的增加而减少。我们还给出了某些长度 $n$ 和维数 $k$ 下具有平凡壳的二元和三元线性码（LCD 码和自正交码）的分类结果，并将所得数量与给定 $n$ 和 $k$ 下所有线性码的数量进行了比较。