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

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