Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tables, bounds and graphics of the smallest known sizes of complete caps
  in the spaces $\mathrm{PG}(3,q)$ and $\mathrm{PG}(4,q)$   In this paper we present and analyze computational results concerning small
complete caps in the projective spaces $\mathrm{PG}(N,q)$ of dimension $N=3$
and $N=4$ over the finite field of order $q$. The results have been obtained
using randomized greedy algorithms and the algorithm with fixed order of points
(FOP). The computations have been done in relatively wide regions of $q$
values; such wide regions are not considered in literature for $N=3,4$. The new
complete caps are the smallest known. Basing on them, we obtained new upper
bounds on $t_2(N,q)$, the minimum size of a complete cap in $\mathrm{PG}(N,q)$,
in particular, \begin{align*} &t_{2}(N,q)<\sqrt{N+2}\cdot
q^{\frac{N-1}{2^{\vphantom{H}}}}\sqrt{\ln q},\quad q\in L_{N},\quad N=3,4,\\
&t_{2}(N,q)<\left(\sqrt{N+1}+\frac{1.3}{\ln
(2q)}\right)q^{\frac{N-1}{2^{\vphantom{H}}}}\sqrt{\ln q},\quad q\in L_{N},\quad
N=3,4, \end{align*} where \begin{align*} &L_{3}:=\{q\le 4673, ~q\
\textrm{prime}\} \cup \{5003,6007,7001,8009\},\\ &L_{4}:=\{q\le 1361, ~q\
\textrm{prime}\} \cup \{1409\}. \end{align*} Our investigations and results
allow to conjecture that these bounds hold for all $q$.

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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph
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The categories of this paper are: hep-ex
3
The categories of this paper are: physics
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The categories of this paper are: patt-sol
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The categories of this paper are: atom-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: math-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: econ
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The categories of this paper are: solv-int
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas