Rate of a binary linear code given all code words have weight integer multiple of 4

Question

Supposed C is a linear binary code with the property that each code word is of Hamming weight n*4 (that is every word has a Hamming weight that is an integer multiple of 4). Show that the rate of C is at most 1/2.

I am stuck on this and mostly looking for a place to start - I know lots of properties for binary linear codes, but I can't seem to relate any of these properties directly to a bound for rate. Any thoughts?

Answer 1

Hint:

1. Show that the code $C$ is self-orthogonal, $C\subseteq C^{\perp}$. Meaning that for all $u,v\in C$ you have $u\perp v$, or $\sum_i u_iv_i=0$.
2. Show that if the length of the code $C$ is $\ell$ and $\dim C=k$, then $\dim C^\perp=\ell-k$.