The hull of a linear code (i.e., a finite field vector space)~\({\mathcal C}\) is defined to be the vector space formed by the intersection of~\({\mathcal C}\) with its dual~\({\mathcal C}^{\perp}.\) Constructing vector spaces with a specified hull dimension has important applications and it is therefore of interest to study minimum distance properties of such spaces. In this paper, we use the probabilistic method to obtain spaces with a given hull dimension and minimum distance and also derive Gilbert-Varshamov type sufficient conditions for their existence.

翻译：线性码（即有限域向量空间）\({\mathcal C}\)的壳被定义为其与对偶码\({\mathcal C}^{\perp}\)的交集所形成的向量空间。构造具有指定壳维数的向量空间具有重要应用，因此研究此类空间的最小距离性质具有重要意义。本文采用概率方法构造了具有给定壳维数和最小距离的空间，并推导了其存在性的Gilbert-Varshamov型充分条件。