Given a [72,36,16] Type II code and a vector of weight 4. You can find
the weight enumerator of its neighbor with respect to this vector using
the Assmus-Mattson theorem and noticing that the vectors of each weight
hold 4-designs. Then
a [68,34,12] code can with the weight enumerator must exist. So if you
can show that a code with the following weight enumerator does not exist
then a [72,36,16] Type II code would not exist.
| Number | Weight |
|---|---|
| 1 | 0 |
| 442 | 12 |
| 14960 | 14 |
| 174471 | 16 |
| 1478048 | 18 |
| 9546537 | 20 |
| 46699952 | 22 |
| 175078410 | 24 |
| 509477760 | 26 |
| 1160564636 | 28 |
| 2081169376 | 30 |
| 2949602799 | 32 |
| 3312254400 | 34 |
| 2949602799 | 36 |
| 2081169376 | 38 |
| 1160564636 | 40 |
| 509477760 | 42 |
| 175078410 | 44 |
| 46699952 | 46 |
| 9546537 | 48 |
| 1478048 | 50 |
| 174471 | 52 |
| 442 | 54 |
| 1 | 68 |
This weight enumerator is given in [1].
[1] S.T. Dougherty and M. Harada, New Extremal Codes of Length 68 ,
IEEE-IT, Vol 46, No. 6, 2133-2136, September 1999.