Complete (k,3)-arcs from quartic curves

Complete (k, 3)-arcs in projective planes over finite fields are the geometric counterpart of linear non-extendible Near MDS codes of length k and dimension 3. A class of infinite families of complete (k, 3)-arcs in PG(2, q) is constructed, for q a power of an odd prime p equivalent to 2(mod 3). The order of magnitude of k is smaller than q. This property significantly distinguishes the complete (k, 3)-arcs of this paper from the previously known infinite families, whose size differs from q by at most 2 root q