A probabilistic construction of small complete caps in projective spaces
Abstract
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. This gives 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 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 $q$.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2014
- DOI:
- 10.48550/arXiv.1406.5060
- arXiv:
- arXiv:1406.5060
- Bibcode:
- 2014arXiv1406.5060B
- Keywords:
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- Mathematics - Combinatorics
- E-Print:
- 32 Pages