In a self dual binary code. Either all codewords have weight divisible by 4 or half do and half dont.

Question

I need to show that in a self dual binary code, either all codewords have weight divisible by 4 or half have weight divisible by 4 and the other half have even weight not divisible by 4.

My attempt has been supposing that not all are divisible by 4. Then because all codewords have even weight in a self dual code, one must be divisible by 2. However, i am not making any progress.

Answer 1

A plan of attack (describe in detail which steps, if any, you have problems with):

1. Show that in a self-orthogonal binary linear code $C$ the congruence $$w(a+b)\equiv w(a)+w(b)\pmod4$$ holds for all words $a,b\in C$.
2. Show that the words $a\in C$ of a weight divisible by four form a subspace $C'\subseteq C$.
3. Show that if $a,b\in C$ both have weights $\equiv2\pmod4$ then $a+b\in C'$.
4. Show that $\dim C'\ge \dim C-1$.

The claim follows from all this.