  For an integer $n\geq 2$, the triangular graph has vertex set the $2$-subsets
of $\{1,\ldots,n\}$ and edge set the pairs of $2$-subsets intersecting at one
point. Such graphs are known to be halved graphs of bipartite rectagraphs,
which are connected triangle-free graphs in which every $2$-path lies in a
unique quadrangle. We refine this result and provide a characterisation of
connected locally triangular graphs as halved graphs of normal quotients of
$n$-cubes. To do so, we study a parameter that generalises the concept of
minimum distance for a binary linear code to arbitrary automorphism groups of
the $n$-cube.
