# 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