  Quadratic residue codes have been one of the most important classes of
algebraic codes. They have been generalized into duadic codes and quadratic
double circulant codes. In this paper we introduce a new subclass of double
circulant codes, called {\em{duadic double circulant codes}}, which is a
generalization of quadratic double circulant codes for prime lengths. This
class generates optimal self-dual codes, optimal linear codes, and linear codes
with the best known parameters in a systematic way. We describe a method to
construct duadic double circulant codes using 4-cyclotomic cosets and give
certain duadic double circulant codes over $\mathbb F_2, \mathbb F_3, \mathbb
F_4, \mathbb F_5$, and $\mathbb F_7$. In particular, we find a new ternary
self-dual $[76,38,18]$ code and easily rediscover optimal binary self-dual
codes with parameters $[66,33,12]$, $[68,34,12]$, $[86,43,16]$, and
$[88,44,16]$ as well as a formally self-dual binary $[82,41,14]$ code.
