  This is a manuscript of a chapter prepared for a book. The good codes possess
large information length and large minimum distance. A class of codes is said
to be asymptotically good if there exists a positive real $\delta$ such that,
for any positive integer $N$ we can find a code in the class with code length
greater than $N$, and with both the rate and the relative minimum distance
greater than $\delta$. The linear codes over any finite field are
asymptotically good. More interestingly, the (asymptotic) GV-bound is a phase
transition point for the linear codes; i.e., asymptotically speaking, the
parameters of most linear codes attain the GV-bound. It is a long-standing open
question: whether or not the cyclic codes over a finite field (which are an
important class of codes) are asymptotically good? However, from a long time
ago the quasi-cyclic codes of index $2$ were proved to be asymptotically good.
This chapter consists of some of our studies on the asymptotic properties of
several classes of quasi-group codes. We'll explain the studies in a consistent
and self-contained style. We begin with the classical results on linear codes.
In many cases we consider the quasi-group codes over finite abelian groups
(including the cyclic case as a subcase of course), and study their asymptotic
properties along two directions: (1) the order of the group (the coindex) is
fixed while the index is going to infinity; (2) the index is small while the
order of the group (the coindex) is going to infinity. Finally we describe the
story on dihedral codes. The dihedral groups are non-abelian but near to cyclic
groups (they have cyclic subgroups of index $2$). The asymptotic goodness of
binary dihedral codes was obtained in the beginning of this century, and
extended to the general dihedral codes recently.
