We study one generator quasi-cyclic codes and four-circulant codes, which are also quasi-cyclic but have two generators. We state the hull dimensions for both classes of codes in terms of the polynomials in their generating elements. We prove results such as the hull dimension of a four-circulant code is even and one-dimensional hull for double-circulant codes, which are special one generator codes, is not possible when the alphabet size $q$ is congruent to 3 mod 4. We also characterize linear complementary pairs among both classes of codes. Computational results on the code families in consideration are provided as well.

翻译：本文研究单生成元准循环码和四循环码（后者虽属准循环码但包含两个生成元）。我们通过生成元中多项式的形式给出这两类码的核维数。证明四循环码的核维数为偶数，且当字母表大小$q$满足模4余3条件时，双循环码（一类特殊的单生成元码）不存在一维核。同时刻画了这两类码中的线性互补对。文中还提供了所考虑码族的相关计算结果。