The weight distribution of the self-dual $[128,64]$ polarity design code

The weight distribution of the binary self-dual $[128,64]$ code being the extended code $C^{*}$ of the code $C$ spanned by the incidence vectors of the blocks of the polarity design in $PG(6,2)$ [11] is computed. It is shown also that $R(3,7)$ and $C^{*}$ have no self-dual $[128,64,d]$ neighbor with $d \in \{ 20, 24 \}$.