  Let $q$ be a prime power; $(q+1,8)$-cages have been constructed as incidence
graphs of a non-degenerate quadric surface in projective 4-space $P(4, q)$. The
first contribution of this paper is a construction of these graphs in an
alternative way by means of an explicit formula using graphical terminology.
Furthermore by removing some specific perfect dominating sets from a
$(q+1,8)$-cage we derive $k$-regular graphs of girth 8 for $k= q-1$ and $k=q$,
having the smallest number of vertices known so far.
