On Complete Arcs Arising from Plane Curves

We point out an interplay between Fq-Frobenius non-classical plane curves and complete (k, d)-arcs in P2(Fq). A typical example that shows how this works is the one concerning an Hermitian curve. We present some other examples here which give rise to the existence of new complete (k, d)-arcs with parameters k = d(q − d + 2) and d = (q − 1)/(q − 1), q being a power of the characteristic. In addition, for q a square, new complete (k, d)-arcs with either k = q √ qb + 1 and d = ( √ q + 1)b (2 ≤ b ≤ √q − 1) or k = (q− 1)√qb+√q +1 and d = (√q +1)b (2 ≤ b ≤ √q− 2) are constructed by using certain reducible plane curves.