# zbMATH — the first resource for mathematics

Extensions of generalized product caps. (English) Zbl 1057.51005
Let $$\text{PG}(n, q)$$ be the projective space of dimension $$n$$ over the finite field $$\mathbb{F}_q$$. A $$k$$-cap $$K$$ in $$\text{PG}(n, q)$$ is a set of $$k$$ points, no three of which are collinear. Let $$m_2(n, q)$$ be the maximum value of $$k$$ for which there exists a $$k$$-cap in $$\text{PG}(n, q)$$ and $$m^{\text{aff}}_2(n, q)$$ be the corresponding value in $$\text{AG}(n, q)$$. The following results are known:
$$m_2(n, 2)= m^{\text{aff}}_2(n, 2)= 2^n$$, $$m_2(2, q)= m^{\text{aff}}_2(2, q)= q +1$$, $$q> 2$$, odd,
$$m_2(2, q)= m^{\text{aff}}_2(2, q)= q+ 2$$, $$q> 2$$ even, and $$m_2(3, q)= q^2+1$$, $$m^{\text{aff}}_2(3, q)= q^2$$, $$q> 2$$.
Besides of these general results are known only the following cases:
$$m_2(4, 3)= m^{\text{aff}}_2(4,3)= 20$$, $$m_2(5, 3)= 56$$, $$m^{\text{aff}}_2(5, 3)= 45$$ and $$m_2(4, 4)= 41$$.
The author of the present article gives some variants of a new construction for caps and obtains improved lower bounds on some values of $$m_2(n, 3)$$. The first examples of improvements are a 1216-cap in $$\text{PG}(9, 3)$$ and a 6464-cap in $$\text{PG}(11, 3)$$. As an application of these construction, the author obtains several caps in ternary affine spaces of larger dimension, which lead to better asymptotics than the caps constructed previously by Calderbank and Fishburn.

##### MSC:
 5.1e+23 Linear codes and caps in Galois spaces
Full Text: