From W. Cary Huffman (2005), On the classification and enumeration of self-dual codes, Finite Fields and Their Applications 11(3) pp 451-490, 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?