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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?

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  • $\begingroup$ @DavidRoberts Weird. I must have done something funny with the code. Thanks for letting me know. $\endgroup$ – Theo Johnson-Freyd Nov 7 '18 at 4:34
  • $\begingroup$ We can delete our comments now :-) $\endgroup$ – David Roberts Nov 7 '18 at 6:03
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I think your question is answered in A Complete Classification of Ternary Self-Dual Codes of Length 24 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.

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  • $\begingroup$ Thanks! For some reason I never found that article when googling. $\endgroup$ – Theo Johnson-Freyd Mar 3 '18 at 1:13
  • $\begingroup$ @TheoJohnson-Freyd No problem! I tried searching “ternary” rather than “type III”, it can be tricky to find things that go by multiple names. $\endgroup$ – John Machacek Mar 3 '18 at 20:45

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