  The bilinear forms graph denoted here by $Bil_q(e\times d)$ is a graph
defined on the set of $(e\times d)$-matrices ($e\geq d$) over $\mathbb{F}_q$
with two matrices being adjacent if and only if the rank of their difference
equals $1$.
  In 1999, K. Metsch showed that the bilinear forms graph $Bil_q(e\times d)$ is
characterized by its intersection array if one of the following holds:
  (-) $q=2$ and $e\geq d+4$,
  (-) $q\geq 3$ and $e\geq d+3$.
  Thus, the following cases have been left unsettled:
  (-) $q=2$ and $e\in \{d,d+1,d+2,d+3\}$,
  (-) $q\geq 3$ and $e\in \{d,d+1,d+2\}$.
  In this work, we show that the graph of bilinear $(d\times d)$-forms over the
binary field, where $d\geq 3$, is characterized by its intersection array. In
doing so, we also classify locally grid graphs whose $\mu$-graphs are hexagons
and the intersection numbers $b_i,c_i$ are well-defined for all $i=0,1,2$.

Combinatorics