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''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c0202702.png" />-cap''
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{{TEX|done}}
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''$k$-cap''
  
A set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c0202703.png" /> points of a finite projective space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c0202704.png" /> no three of which are collinear. Two caps are considered equivalent if there is a collineation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c0202705.png" /> transforming one into the other. The search for the maximal number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c0202706.png" /> of points of a cap in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c0202707.png" />, the construction, and the classification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c0202708.png" />-caps form important problems in the study of caps that are not yet (1984) completely solved. The following results are known (see [[#References|[2]]], [[#References|[3]]]):
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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 [[#References|[2]]], [[#References|[3]]]):
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c0202709.png" />; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027010.png" />-cap is unique (up to equivalence) and is a set of points not located in a fixed hyperplane in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027011.png" />;
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$m(n,2)=2^n$; the $2^n$-cap is unique (up to equivalence) and is a set of points not located in a fixed hyperplane in $P(n,2)$;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027012.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027013.png" /> is odd; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027014.png" />-cap in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027015.png" /> is unique and is a conic;
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$m(2,q)=q+1$ if $q$ is odd; the $(q+1)$-cap in $P(2,q)$ is unique and is a conic;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027016.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027017.png" /> is even; a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027018.png" />-cap in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027019.png" /> is, generally speaking, not unique;
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$m(2,q)=q+2$ if $q$ is even; a $(q+2)$-cap in $P(2,q)$ is, generally speaking, not unique;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027020.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027021.png" /> is odd the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027022.png" />-cap in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027023.png" /> is unique and is an elliptic quadric; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027024.png" /> is even it is, generally speaking, not unique;
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$m(3,q)=q^2+1$. 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;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027025.png" />; a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027026.png" />-cap in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027027.png" /> is not unique;
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$m(4,3)=20$; a $20$-cap in $P(4,3)$ is not unique;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027028.png" />; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027029.png" />-cap in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027030.png" /> is unique.
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$m(5,3)=56$; the $56$-cap in $P(5,3)$ is unique.
  
 
Caps are used in coding theory (cf., e.g., [[#References|[2]]]).
 
Caps are used in coding theory (cf., e.g., [[#References|[2]]]).
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====Comments====
 
====Comments====
In the (differential) topology of surfaces a cap of the second kind or cross cap is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020270/c02027031.png" />-dimensional manifold with boundary homeomorphic to the [[Möbius strip|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|Theory of surfaces]] for more details.
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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|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|Theory of surfaces]] for more details.

Latest revision as of 08:46, 1 August 2014

$k$-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.

How to Cite This Entry:
Cap. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cap&oldid=32636
This article was adapted from an original article by V.V. Afanas'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article