Asymptotic performance of double and four circulant codes with small hull dimension

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.