A 3-cycle construction of complete arcs sharing $(q+3)/2$ points with a conic

In the projective plane PG(2, q), q equivalent to 2 ( mod 3) odd prime power, q >= 11, an explicit construction of 1/2(q + 7)-arcs sharing 1/2(q + 3) points with an irreducible conic is considered. The construction is based on 3-orbits of some projectivity, called 3-cycles. For every q, variants of the construction give non-equivalent arcs. It allows us to obtain complete 1/2(q + 7)-arcs for q <= 4523. Moreover, for q = 17,59 there exist variants that are incomplete arcs. Completing these variants we obtained complete (1/2(q+3) + delta)-arcs with delta = 4, q = 17, and delta = 3, q = 59; a description of them as union of some symmetrical objects is given