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

TL;DR

This paper introduces new inductive methods to construct small complete caps in projective spaces over even finite fields, significantly improving upper bounds for their minimal sizes in higher dimensions.

Contribution

It presents novel inductive constructions for complete caps in PG(N,q) with q even, reducing the problem to the plane and improving known bounds for large q.

Findings

Improved upper bounds for smallest complete caps in PG(N,q) for q ≥ 2^3

Construction methods applicable to infinite q square values and specific q ranges

Enhanced bounds for dimensions N ≥ 4 and various q values

Abstract

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 $q\leq 2^{18}.$