Recall that a *binary code* is a subgroup $C \subset \mathbb F_2^n$; the elements of $C$ are called *code words*. The *Hamming weight* of a code word $c\in C$ is the number of $1$s in it. A binary code is *self-dual* if $C = C^\perp := \{v \in \mathbb F_2^n : \langle v,c\rangle = 0\in \mathbb F_2\}$. Self-dual codes automatically satisfy that all code words have Hamming weight divisible by $2$ (since every element must be orthogonal to itself).

There are many applications for binary codes in which all Hamming weights are divisible by $4$. A famous theorem says that a self-dual code of this type can exist only when $n$ is a multiple of $8$.

Do there exist self-dual codes all of whose Hamming weights are divisible by $8$? What is the smallest one? What dimensions do they exist in?

Note that the binary Golay code, related to the famous Leech lattice, is not of this type, since it has code words with Hamming weight $12$.