Recall that a *Type III code* of rank $r$ is a linear subspace $C \subset \mathbb F_3^r$ which is self-dual for the standard inner product. (These occur only when $r$ is divisible by $4$.) Elements of $C$ are called *code words*. The *Hamming weight* of a code word is its number of non-zero entries. I will call a code word *maximal* if its Hamming weight is $r$. (Maximal codewords clearly only occur when $r$ is divisible by $12$.)

The set of maximal codewords can be partitioned into two subsets as follows: two maximal words $w_1,w_2$ are in the same subset if $w_1-w_2$ has even Hamming weight, and are in opposite subsets if $w_1-w_2$ has odd Hamming weight. Call these subsets $M_+$ and $M_-$.

I suggest the following definition:

**Definition:** The *index* of $C$ is $|M_+| - |M_-|$. Of course, this is only defined up to sign. I could take its absolute value if I cared. The name is because this is related to the "supersymmetric index" of a certain supersymmetric field theory.

The discussion in the comments of this question implies the following when $r$ is divisible by $24$:

**Proposition:** The index of any Type III code of rank $r=24k$ is divisible by $24$.

**Conjecture:** This is true also when $r \equiv 12 \mod 24$. (It is trivially true for $r \equiv \pm 4 \mod 12$, as then $M_\pm$ are empty.)

Moreover, I suspect that both $|M_+|$ and $|M_-|$ are necessarily divisible by $24$.

**Example:** The Ternary Golay code, with rank $12$ has index $24$. It is the unique rank-$12$ code with non-zero index.

**Example:** The complete classification of Type III codes of rank $24$ is known. Assuming I read it correctly, there are precisely five codes of rank $24$ and index $24$.

Question:In higher rank, do there exist Type III codes with index exactly $24$? For example, what about rank $36$?

At best, there would be some general algorithm that produces a code with index $24$ for each rank $r = 12k$.

By the way, I know how to prove:

**Proposition:** If the code $C$ contains words of Hamming weight $3$, then its index vanishes.

So if you are looking for such a code, you know not to look at such codes.

It's pretty easy to show:

**Lemma:** The index multiplies when you take the direct sum of codes.

Since $24^2$ is pretty big, you probably will want to work with indecomposable codes.