Binary linear codes and weights Let $C$ be a binary $[n,k]$ linear code. If $C$ has a codeword of odd weight
then the codewords of even weightin $C$ form an $[n,k-1]$ binary linear code. Can you prove that?
Hint: Consider the linear mapping $f$ from $C$ to $GF(2)$ defined by 
$$
f:(a_1,a_2,\ldots,a_n)\mapsto a_1+a_2+\cdots+a_n.
$$ 
Show that this is onto. Apply the rank-nullity theorem (familiar from linear algebra) to it.
Let x be an odd codeword so that the cosets x+E and E are distinct. Given any coset y+E we must have y+E=E if y is even. Otherwise y is odd while x+y is even implying x+E = y+E. Therefore, C/E has size 2 and dimension 1. Therefore dim(E) = dim(C) - 1.