Self-Dual Double Circulant, Self-Dual Double Negacirculant and LCD Double Negacirculant Codes Over the Ring Fq[u,v]/2 - u, v2-v, uv-vu>

In this paper, we investigate self-dual double circulant, and self-dual and linear complementary dual (LCD) double negacirculant codes over a finite ring
$R = \mathbb F_{q} + u \mathbb F_{q} + v \mathbb F_{q} + uv\mathbb F_{q}$
, where
$u^{2}=u$
,
$v^{2}=v$
,
$uv=vu $
and
$q=p^{m}$
. We study the algebraic structure of double circulant codes over
$R$
. We provide necessary and sufficient conditions for a double circulant code to be a self-dual code. We give a formula to get the total number of self-dual double circulant codes over the ring
$R$
. We compute distance bounds for self-dual double circulant codes over
$R$
. In addition, by using a Gray map, we show that the families of self-dual double circulant codes under the Gray map are asymptotically good. Moreover, the algebraic structure of double negacirculant codes and necessary and sufficient conditions for a double negacirculant code to be a self-dual code and to be an LCD code are also given. We determine the total number of self-dual and LCD double negacirculant codes over
$R$
.