A construction of small complete caps in projective spaces

In this work complete caps in PG(N, q) of size $${O(q^{\frac{N-1}{2}} \log^{300} q)}$$O(qN-12log300q) are obtained by probabilistic methods. This gives an upper bound asymptotically very close to the trivial lower bound $${\sqrt{2}q^{\frac{N-1}{2}}}$$2qN-12 and it improves the best known bound in the literature for small complete caps in projective spaces of any dimension. The result 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.