In the projective planes PG(2, q), with the help of a computer search using randomized greedy algorithms, more than 5030 new small complete arcs are obtained for q ∈ H = H_1 ∪ H_2 ∪ S where H_1 = {q : 19 ≤ q ≤ 44519, q is a power prime}, H_2 = {q : 44531 ≤ q ≤ 66749, q prime}, S is a set of 110 sporadic primes q with 67003 ≤ q ≤ 300007. Using the new arcs, it is shown that for the smallest size t_2(2, q) of a complete arc in PG(2, q), q ∈ H, it holds that t_2(2, q) < D√q(ln q)f(q;D) where f(q;D) is a decreasing function of q, D is a con- stant independent of q, and f(q; 0.6) = 1.51/ ln q + 0.8028. Moreover, our results allow us to conjecture that all above mentioned upper estimates hold for all q ≥ 19.
Two types of upper bounds on the smallest size of a complete arc in the plane PG(2, q)
BARTOLI, DANIELE;FAINA, Giorgio;MARCUGINI, Stefano;PAMBIANCO, Fernanda
2013
Abstract
In the projective planes PG(2, q), with the help of a computer search using randomized greedy algorithms, more than 5030 new small complete arcs are obtained for q ∈ H = H_1 ∪ H_2 ∪ S where H_1 = {q : 19 ≤ q ≤ 44519, q is a power prime}, H_2 = {q : 44531 ≤ q ≤ 66749, q prime}, S is a set of 110 sporadic primes q with 67003 ≤ q ≤ 300007. Using the new arcs, it is shown that for the smallest size t_2(2, q) of a complete arc in PG(2, q), q ∈ H, it holds that t_2(2, q) < D√q(ln q)f(q;D) where f(q;D) is a decreasing function of q, D is a con- stant independent of q, and f(q; 0.6) = 1.51/ ln q + 0.8028. Moreover, our results allow us to conjecture that all above mentioned upper estimates hold for all q ≥ 19.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
