Set $ N_c (i,4)$ as the number of codewords in a binary code C with hamming weights congruent to $i$ module $4$. Assuming C is of length $n$ and dimension $k$, how can it be proved that if $C$ is an $[n,k]$-code with $k \ge 4 $ that $ N_c (i,4)$ is even $\forall i \in0,1,2,3 $.
i know that $N_c (0,4) + N_c (2,4)$ is the sum of all even weighted elements and $N_c (1,4) + N_c (3,4)$ is the sum of all odd weighted elements. I was planning to perform an induction on $n$ starting with $k = 4$ but i dont know how to continue.Any suggestions? thanks a great deal.
By the way, is it safe for me to assume that either $ \{N_c (0,4) + N_c (2,4)\} = |C|$ or $N_c (0,4) + N_c (2,4) = N_c (1,4) + N_c (3,4) = 2^{k-1} ?$ - with no restrictions on $k$