Abstract In this paper, we give a geometric construction of the three strong non-lifted $$(3\mod 5)$$ ( 3 mod 5 ) -arcs in $${{\,\textrm{PG}\,}}(3,5)$$ PG ( 3 , 5 ) of respective sizes 128, 143, and 168, and construct an infinite family of non-lifted, strong $$(t\mod q)$$ ( t mod q ) -arcs in $${{\,\textrm{PG}\,}}(r,q)$$ PG ( r , q ) with $$t=(q+1)/2$$ t = ( q + 1 ) / 2 for all $$r\ge 3$$ r ≥ 3 and all odd prime powers q.