  We consider the bipartite version of the {\it degree/diameter problem},
namely, given natural numbers $d\ge2$ and $D\ge2$, find the maximum number
$\N^b(d,D)$ of vertices in a bipartite graph of maximum degree $d$ and diameter
$D$. In this context, the bipartite Moore bound $\M^b(d,D)$ represents a
general upper bound for $\N^b(d,D)$. Bipartite graphs of order $\M^b(d,D)$ are
very rare, and determining $\N^b(d,D)$ still remains an open problem for most
$(d,D)$ pairs.
  This paper is a follow-up to our earlier paper \cite{FPV12}, where a study on
bipartite $(d,D,-4)$-graphs (that is, bipartite graphs of order $\M^b(d,D)-4$)
was carried out. Here we first present some structural properties of bipartite
$(d,3,-4)$-graphs, and later prove there are no bipartite $(7,3,-4)$-graphs.
This result implies that the known bipartite $(7,3,-6)$-graph is optimal, and
therefore $\N^b(7,3)=80$. Our approach also bears a proof of the uniqueness of
the known bipartite $(5,3,-4)$-graph, and the non-existence of bipartite
$(6,3,-4)$-graphs.
  In addition, we discover three new largest known bipartite (and also
vertex-transitive) graphs of degree 11, diameter 3 and order 190, result which
improves by 4 vertices the previous lower bound for $\N^b(11,3)$.
