I am studying error correcting codes and their properties. A Type II code is a linear subspace $C \subset \mathbb{F}_2^n$ of dimension $n/2$ which satisfies:
1: $C=C^*=\{x\in \mathbb{F}_2^n :\sum x_i u_i \equiv 0 \ \forall u \in C \}$
2: Each codeword $u \in C$ has a doubly even number of nonzero entries.
I can show using weight enumerators that a Type II code can only exist if $8|n$, but it is stated in many books that a Type II code of length $n$ exists if and only if $8|n$. I can prove that a quadratic residue code of length $8n-1$ where $8n-1$ is prime produces a Type II code but this clearly doesn't work for all $n$. What proves the other half of this statement?
There is a type II code of length $8$ (the extended Hamming code). Simply take the direct sum of $k$ copies of this to get a type II code of length $8k$. In detail, the words in this codes are the concatenations of $k$ words each drawn from the extended Hamming code.