Let \(\text{PG}(2, q)\) be the projective plane over the Galois field \(\text{GF}(q)\). A \(k\)-arc in \(\text{PG}(2, q)\) is a set of \(k\) points, no three of which are collinear. A \(k\)-arc is called complete if it is not contained in a \((k + 1)\)-arc of \(\text{PG}(2, q)\). The authors use the following notations in \(\text{PG}(2, q): m_2(2, q)\) is the size of the largest complete arc, \(m_2'(2, q)\) is the size of the second largest complete arc, and \(t_2(2, q)\) is the size of the smallest complete arc. The corresponding best known values are denoted by \(\overline m_2'(2, q)\) and \(\overline t_2(2, q)\).
In the present article a number of new values of \(\overline m_2'(2, q)\) and \(\overline t_2(2, q)\) are obtained by computer search. Many new sizes \(k\) for which a complete \(k\)-arc in \(\text{PG}(2, q)\) exists are also obtained. New upper bounds on the smallest size \(t_2(2, q)\) complete are obtained for \(q= 41,43,47,49,53, 59,64,\) \(71\leq q\leq 809\), \(q\neq 529,625,729\) and \(q= 821\). These new upper bounds give \(t_2(2, q)< 4\sqrt{q}\) for \(3\leq q\leq 809\) and \(q= 821\). New lower bound on the second largest size \(m_2'(2, q)\) of a complete are obtained for \(q= 31,41,43,47,53, 125\). These lower bounds are as follows: \(m_2'(2,31)\geq 22\), \(m_2'(2, 41)\geq 30\), \(m_2'(2, 43)\geq 28\), \(m_2'(2, 47)\geq 32\), \(m_2'(2, 53)\geq 42\), \(m_2'(2, 125)\geq 66\).