Non-existence of sets of type $$\mathbf (0, 1, 2 , {\varvec{n}}_{{\varvec{d}}})_{{\varvec{d}}}$$ ( 0 , 1 , 2 , n d ) d in PG( $${\varvec{r,q}}$$ r , q ) with $$\mathbf 3 \le {\varvec{d}}\le {\varvec{r}}-\mathbf 1 $$ 3 ≤ d ≤ r - 1 and $${\varvec{r}}\ge \mathbf 4 $$ r ≥ 4