2014/04/02 by Daniele Bartoli, Bartoli, Daniele, Alexander A. Davydov +9
Mathematics · Engineering · Computer Science · #Finite Group Theory Research #graph theory and CDMA systems #Coding theory and cryptography
paper · pdf · doi:10.48550/arxiv.1404.0469
In the previous works of the authors, a step-by-step algorithm FOP which uses\nany fixed order of points in the projective plane \PG(2,q) is\nproposed to construct small complete arcs. In each step, the algorithm adds to\na current arc the first point in the fixed order not lying on the bisecants of\nthe arc. The algorithm is based on the intuitive postulate that\n\PG(2,q) contains a sufficient number of relatively small complete\narcs. Also, in the previous papers, it is shown that the type of order on the\npoints of \PG(2,q) is not relevant. A complete lexiarc in\n\PG(2,q) is a complete arc obtained by the algorithm FOP using the\nlexicographical order of points. In this work, we collect and analyze the sizes\nof complete lexiarcs in the following regions: \amp;
textbfall \nq
le321007,~ q
mbox prime power; amp; 15
mbox sporadic q's in the interval\n[323761
ldots430007],
mbox see (1.10). In the work [9], the\nsmallest known sizes of complete arcs in \PG(2,q) are collected for\nall q\≤160001, q prime power. The sizes of complete arcs, collected in\nthis work and in [9], provide the following upper bounds on the smallest size\nt2(2,q) of a complete arc in the projective plane \PG(2,q):\n\t2(2,q)amp;lt;0.998
sqrt3q
ln qlt;1.729
sqrtq
ln qamp;
mbox for\namp;amp;7amp;
le q
le160001;
t2(2,q)amp;lt;1.05
sqrt3q
ln qlt;1.819
sqrtq
ln qamp;
mbox\nfor amp;amp;7amp;
le q
le321007. Our investigations and results allow to\nconjecture that the bound t2(2,q)<1.05\√(3q\ln q)<1.819\√(q\ln q)\nholds for all q\≥7. It is noted that sizes of the random complete arcs and\ncomplete lexiarcs behave similarly. This work can be considered as a\ncontinuation and development of the paper [11].\n