  For every linear binary code $C$, we construct a geometric triangular
configuration $\Delta$ so that the weight enumerator of $C$ is obtained by a
simple formula from the weight enumerator of the cycle space of $\Delta$. The
triangular configuration $\Delta$ thus provides a geometric representation of
$C$ which carries its weight enumerator. This is the initial step in the
suggestion by M. Loebl, to extend the theory of Pfaffian orientations from
graphs to general linear binary codes. Then we carry out also the second step
by constructing, for every triangular configuration $\Delta$, a triangular
configuration $\Delta'$ and a bijection between the cycle space of $\Delta$ and
the set of the perfect matchings of $\Delta'$.
