Cap
-cap
A set of points of a finite projective space
no three of which are collinear. Two caps are considered equivalent if there is a collineation of
transforming one into the other. The search for the maximal number
of points of a cap in
, the construction, and the classification of
-caps form important problems in the study of caps that are not yet (1984) completely solved. The following results are known (see [2], [3]):
; the
-cap is unique (up to equivalence) and is a set of points not located in a fixed hyperplane in
;
if
is odd; the
-cap in
is unique and is a conic;
if
is even; a
-cap in
is, generally speaking, not unique;
. If
is odd the
-cap in
is unique and is an elliptic quadric; if
is even it is, generally speaking, not unique;
; a
-cap in
is not unique;
; the
-cap in
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 -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.
Cap. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cap&oldid=32636