Even poset and a parity result for binary linear code

Let C be a linear binary code, namely a subspace of the space consisting of all binary vectors of a fixed length. A vector in C is maximal provided it has a maximal support among C; a nonzero vector in C is minimal provided it has a minimal support among C⧹{0}. We prove that the sum of all maximal vectors of C equals the sum of all minimal vectors of C. In course of this research, we introduce the concept of even poset and establish a duality result for it.