From [W. Cary Huffman (2005), On the classification and enumeration of self-dual codes, Finite Fields and Their Applications 
11(3) pp 451-490](https://www.sciencedirect.com/science/article/pii/S1071579705000444), I learn that there are at least 140 Type III codes of length 24, of which two have the property that all code words have Hamming length at least 9. (One of these is a quadratic residue code. I don't know the other one.) 9 is the maximal minimal-Hamming-length possible for any Type III code of length 24.


How many of Type III codes of length 24 have Hamming lengths bounded by 6? I.e. how many do not have words of Hamming length 3?



How many length-24 Type III codes have no words of Hamming weight 3?
I think your question is answered in [A Complete Classification of Ternary Self-Dual Codes of Length 24](https://arxiv.org/abs/0804.0637) by Harada and Munemasa. Theorem 1 of the paper claims there are 166 inequivalent ternary self dual codes of weight 6 and 170 inequivalent ternary self dual codes of weight 3.


