An explicit formula for obtaining $(q+1,8)$-cages and others small   regular graphs of girth 8
  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. 
M. Abreu, G. Araujo-Pardo, C. Balbuena, D. Labbate
math.CO