  We study the asymptotic behavior of double and four circulant codes, which
are quasi-cyclic codes of index two and four respectively. Exact enumeration
results are derived for these families of codes with the prescribed hull
dimension. These formulas, in turn, are the most used tools to prove the good
behavior of double circulant and four circulant codes asymptotically.
Computational results on the code families in consideration are provided as
well.
