Complete (k,4) -arcs from quintic curves

Complete (k, 4)-arcs in projective Galois planes are the geometric counterpart of linear non-extendible codes of length k, dimension 3 and Singleton defect 2. A class of infinite families of complete (k, 4)-arcs in PG (2 , q) is constructed, for q a power of an odd prime p≡3(mod4), p> 3. The order of magnitude of k is smaller than q. This property significantly distinguishes the complete (k, 4)-arcs of this paper from the previously known infinite families, whose size exceeds q-6q.