On the hull and complementarity of one generator quasi-cyclic codes and four-circulant codes

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.