Search results
Results from the WOW.Com Content Network
A projective plane of order N is a Steiner S(2, N + 1, N 2 + N + 1) system (see Steiner system). Conversely, one can prove that all Steiner systems of this form (λ = 2) are projective planes. The number of mutually orthogonal Latin squares of order N is at most N − 1. N − 1 exist if and only if there is a projective plane of order N.
Edward Assmus presented a connection between projective planes and coding theory at the conference Combinatorial Aspects of Finite Geometries in 1970. [4] He studied the code generated by the rows of the incidence matrix of a hypothetical projective plane of order ten and derived a number of restrictive properties that such a code must satisfy.
If P is a finite set, the projective plane is referred to as a finite projective plane. The order of a finite projective plane is n = k – 1, that is, one less than the number of points on a line. All known projective planes have orders that are prime powers. A projective plane of order n is an ((n 2 + n + 1) n + 1) configuration. The smallest ...
The quotient map from the sphere onto the real projective plane is in fact a two sheeted (i.e. two-to-one) covering map. It follows that the fundamental group of the real projective plane is the cyclic group of order 2; i.e., integers modulo 2.
The theorem, for example, rules out the existence of projective planes of orders 6 and 14 but allows the existence of planes of orders 10 and 12. Since a projective plane of order 10 has been shown not to exist using a combination of coding theory and large-scale computer search, [1] the condition of the theorem is evidently not sufficient for ...
In 1992 he received the Lester Randolph Ford Award for the article The search for a finite projective plane of order 10. [3] The eponymous Lam's problem is equivalent to finding a finite projective plane of order 10 or finding 9 orthogonal Latin squares of order 10. [4]
If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2. The Fano plane is called the projective plane of order 2 because it is unique (up to isomorphism). In general, the projective plane of order n has n 2 + n + 1 points and the same number of lines; each line ...
Any finite projective plane of order n is an ((n 2 + n + 1) n + 1) configuration. Let Π be a projective plane of order n. Remove from Π a point P and all the lines of Π which pass through P (but not the points which lie on those lines except for P) and remove a line ℓ not passing through P and all the points that are on line ℓ.