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.

翻译：我们研究了双重循环码和四重循环码的渐近行为，它们分别为指标为2和4的准循环码。针对具有指定壳维数的这类码族，推导出了精确计数结果。这些公式是证明双重循环码和四重循环码在渐近意义上具有良好性质的最常用工具。此外，还给出了所考虑码族的计算结果。