New inductive constructions of complete caps in $PG(N,q)$, $q$ even

Details New inductive constructions of complete caps in $PG(N,q)$, $q$ even New inductive constructions of complete caps in $PG(N,q)$, $q$ even

2009-01-04

arxiv.org/abs/0901.0367v1

Mathematics

Combinatorics

Scientific paper

Some new families of small complete caps in $PG(N,q)$, $q$ even, are described. By using inductive arguments, the problem of the construction of small complete caps in projective spaces of arbitrary dimensions is reduced to the same problem in the plane. The caps constructed in this paper provide an improvement on the currently known upper bounds on the size of the smallest complete cap in $PG(N,q),$ $N\geq 4,$ for all $q\geq 2^{3}.$ In particular, substantial improvements are obtained for infinite values of $q$ square, including $ q=2^{2Cm},$ $C\geq 5,$ $m\geq 3;$ for $q=2^{Cm},$ $C\geq 5,$ $m\geq 9,$ with $C,m$ odd; and for all $q\leq 2^{18}.$

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