Cap

$k$-cap

A set of $k$ points of a finite projective space $P(n,q)$ no three of which are collinear. Two caps are considered equivalent if there is a collineation of $P(n,q)$ transforming one into the other. The search for the maximal number $m(n,q)$ of points of a cap in $P(n,q)$, the construction, and the classification of $m(n,q)$-caps form important problems in the study of caps that are not yet (1984) completely solved. The following results are known (see [2], [3]):

$m(n,2)=2^n$; the $2^n$-cap is unique (up to equivalence) and is a set of points not located in a fixed hyperplane in $P(n,2)$;

$m(2,q)=q+1$ if $q$ is odd; the $(q+1)$-cap in $P(2,q)$ is unique and is a conic;

$m(2,q)=q+2$ if $q$ is even; a $(q+2)$-cap in $P(2,q)$ is, generally speaking, not unique;

$m(3,q)=q^2+1$. If $q$ is odd the $(q^2+1)$-cap in $P(3,q)$ is unique and is an elliptic quadric; if $q$ is even it is, generally speaking, not unique;

$m(4,3)=20$; a $20$-cap in $P(4,3)$ is not unique;

$m(5,3)=56$; the $56$-cap in $P(5,3)$ is unique.

Caps are used in coding theory (cf., e.g., [2]).

References

[1]	R.C. Bose, "Mathematical theory of the symmetrical factorial design" Shankhyā , 8 (1947) pp. 107–166
[2]	R. Hill, "Caps and codes" Discrete Math. , 22 : 2 (1978) pp. 111–137
[3]	B. Segre, "Introduction to Galois geometries" Atti Accad. Naz. Lincei Mem. , 8 (1967) pp. 133–236

Comments

In the (differential) topology of surfaces a cap of the second kind or cross cap is a $2$-dimensional manifold with boundary homeomorphic to the Möbius strip used to construct (more complicated) surfaces by removing a disc and replacing it with a cross cap or with a bundle (also called a cap of the first kind). Cf. Theory of surfaces for more details.