  In this paper, we study a special kind of factorization of $x^n+1$ over
$\mathbb{F}_q, $ with $q$ a prime power $\equiv 3~({\rm mod}~4)$ when $n=2p,$
with $p\equiv 3~({\rm mod}~4)$ and $p$ is a prime. Given such a $q$ infinitely
many such $p$'s exist that admit $q$ as a primitive root by the Artin
conjecture in arithmetic progressions. This number theory conjecture is known
to hold under GRH. We study the double (resp. four)-negacirculant codes over
finite fields $\mathbb{F}_q, $ of co-index such $n$'s, including the exact
enumeration of the self-dual subclass, and a modified Varshamov-Gilbert bound
on the relative distance of the codes it contains.
