  A connected graph K has a modulo 2 embedding to the sphere with g handles if
and only if there is a general position PL map f of K in the plane and a
symmetric square matrix A of size |E(K)| with modulo 2 entries and zeros on the
diagonal such that $rk A \leq 2g$ and $A_{\sigma, \tau} = |f\sigma \cap f\tau|
mod 2$ for any non-adjacent edges $\sigma, \tau$. This is essentially proved by
R. Fulek and J. Kyn\v{c}l. The main of results of this note is an alternative
proof of the (=>) part.
