A probabilistic construction of small complete caps in projective spaces

In this work complete caps in PG(N,q) of size O(q N−1 2 log 300 q) are obtained by probabilistic methods. This gives an upper bound asymptotically very close to the trivial lower bound √ 2q N−1 2 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.

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