Even poset and a parity result for binary linear code

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