Maximal (n,3)-arcs in PG(2,13)

In this paper we show, using a computer-based search exploiting relations of inclusion between arcs and (n, 3)-arcs and projective equivalence properties, that the largest size of a complete (n, 3)-arc in PG(2, 13) is 23 and that only seven non-equivalent (23, 3)-arcs exist. From this result, we deduce the non-existence of some [n, k, n−k]13 linear codes and bounds on the minimum distance of some [n, 3, d]13 linear codes. Moreover, we determine the spectrum of the sizes of the complete (n, 3)-arcs in PG(2, 13) and the classification of the smallest complete (n, 3)-arcs.