 Even-weight subcode of a binary linear code

For a binary linear code $C$, the linear functional $\sigma (x)={\sum }_j{x}_j$ records Hamming weight modulo two. Its kernel of a linear map is the even-weight subcode. If $\sigma {|}_C$ vanishes, this is all of $C$; otherwise rank-nullity theorem gives dimension $\dim C-1$. This also proves closure under addition without treating ordinary integer weight as a linear function.