The Geometry of $(t\mathrm{m}\mathrm{o}\mathrm{d}q)$-arcs

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