Geometric representation of binary codes and computation of weight enumerators

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 $\Delta'$.