ON THE STRUCTURE OF (t mod q)-ARCS IN FINITE PROJECTIVE GEOMETRIES

In this paper, we introduce constructions and structure results for (t mod q)-arcs. We prove that all (2 mod q)-arcs in PG(r, q) with $r \geq  3$ are lifted. We find all (3 mod 5) plane arcs of small cardinality not exceeding 33 and prove that every (3mod 5)-arc in PG(3, 5) of size at most 158 is lifted. This result is applied further to rule out the existence of (104, 22)-arcs in PG(3, 5) which solves an open problem on the optimal size of fourdimensional linear codes over $\mathbb{F}_5$.