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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.

51E22 Linear codes and caps in Galois spaces
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