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In this work complete caps in $PG(N,q)$ of size$O(q^{\frac{N-1}{2}}\log^{300} q)$ are obtained by probabilistic methods. Thisgives an upper bound asymptotically very close to the trivial lower bound$\sqrt{2}q^{\frac{N-1}{2}}$ and it improves the best known bound in theliterature for small complete caps in projective spaces of any dimension. Theresult obtained in the paper also gives a new upper bound for $l(m,2,q)_4$,that is the minimal length $n$ for which there exists an $[n,n-m, 4]_q2$covering code with given $m$ and $q$.

A probabilistic construction of small complete caps in projective spaces