On the hull and complementarity of 1-generator quasi-cyclic codes and four-circulant codes

We study 1-generator quasi-cyclic codes and four-circulant codes, which are also quasi-cyclic but have 2 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 1-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.