A class of self-dual quasi-abelian codes of index 2 over any finite field $F$ is introduced. By counting the number of such codes and the number of the codes in this class whose relative minimum weights are small, such codes are proved to be asymptotically good provided −1 is a square in $F$ . Moreover, a class of self-orthogonal quasi-abelian codes of index 2 is defined; and such codes always exist. In a way similar to that for self-dual quasi-abelian codes of index 2, it is proved that these self-orthogonal quasi-abelian codes of index 2 are asymptotically good.

中文翻译：

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自对偶 2-拟阿贝尔码

任意有限域上索引为 2 的一类自对偶拟阿贝尔码 $F$介绍。通过计算此类代码的数量和该类中相对最小权重较小的代码数量，证明此类代码是渐近好的，前提是 -1 是 $F$ . 此外，定义了一类索引为2的自正交拟阿贝尔码；并且这样的代码总是存在的。以类似于索引2的自对偶准阿贝尔码的方式，证明了这些索引2的自正交准阿贝尔码是渐近良好的。