Tables, bounds and graphics of the smallest known sizes of complete arcs in the plane for all and sporadic in the interval
arXiv:1312.2155
Abstract
In the projective planes , we collect the smallest known sizes of complete arcs for the regions \begin{align*} &\mbox{all } q\le160001,~~ q \mbox{ prime power};\\ &Q_{4}=\{34 \mbox{ sporadic }q'\mbox{s in the interval }[160801\ldots430007], \mbox{ see Table 3}\}. \end{align*} For , the collection of arc sizes is complete in the sense that arcs for all prime powers are considered. This proves new upper bounds on the smallest size of a complete arc in , in particular \begin{align*} t_{2}(2,q)&<0.998\sqrt{3q\ln q}<1.729\sqrt{q\ln q}&\mbox{ for }&&7&\le q\le160001;~~(1) \\ t_{2}(2,q)&<\sqrt{q}\ln^{0.7295}q&\mbox{ for }&&109&\le q\le160001;~~(2)\\ t_{2}(2,q)&<\sqrt{q}\ln^{c_{up}(q)}q,~~c_{up}(q)=\frac{0.27}{\ln q}+0.7,&\mbox{ for }&&19&\le q\le160001;~~(3)\\ t_{2}(2,q)&<0.6\sqrt{q}\ln^{φ_{up}(q;0.6)} q,~~φ_{up}(q;0.6)=\frac{1.5}{\ln q}+0.802,&\mbox{ for }&&19&\le q\le160001.~~(4) \end{align*} Moreover, the bounds (2) -- (4) hold also for . Also, \begin{align*} t_{2}(2,q)&<1.006\sqrt{3q\ln q}<1.743\sqrt{q\ln q}&\mbox{ for }&&q\in Q_{4}.~~(5) \end{align*} Our investigations and results allow to conjecture that the bounds (2) -- (5) hold for all .
85 pages, 11 figures, 6 tables, 83 references; data and figures are updated, the region of data is increased; the title is changed; figures and references are added