Open problems in finite projective spaces
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Most of the objects studied in this paper have an interesting group; the classical groups and other finite simple groups appear in this way.About:
This article is published in Finite Fields and Their Applications.The article was published on 2015-03-01 and is currently open access. It has received 40 citations till now. The article focuses on the topics: Finite geometry & Galois geometry.read more
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On planes through points off the twisted cubic in PG(3,q) and multiple covering codes
TL;DR: The structure of the point-plane incidence matrix in PG ( 3 , q ) with respect to the orbits of points and planes under the action of the stabilizer group of the twisted cubic is described to view generalized doubly-extended Reed-Solomon codes of codimension four as asymptotically optimal multiple covering codes.
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On cosets weight distributions of the doubly-extended Reed-Solomon codes of codimension 4
TL;DR: It is proved that the difference between the w-th components of the distributions is uniquely determined by the Difference between the 3-rd components, which implies an interesting (and in some sense unexpected) symmetry of the obtained distributions.
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Upper bounds on the smallest size of a complete arc in a finite Desarguesian projective plane based on computer search
Daniele Bartoli,Alexander A. Davydov,Giorgio Faina,Alexey Kreshchuk,Stefano Marcugini,Fernanda Pambianco +5 more
TL;DR: In this article, the results of computer search obtained and collected in the previous works of the authors and in the present paper are investigated for the projective plane PG(2, q) in particular for a wide region of values of q, and the search is complete in the sense that all prime powers are considered.
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Twisted cubic and point-line incidence matrix in $$\mathrm {PG}(3,q)$$
TL;DR: In this article, the authors considered the structure of the point-line incidence matrix of the projective space, which is connected with orbits of points and lines under the stabilizer group of the twisted cubic and investigated structures of submatrices with incidences between a union of line orbits and an orbit of points.
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On Cosets Weight Distribution of Doubly-Extended Reed-Solomon Codes of Codimension 4
TL;DR: In this article, the authors considered the generalized doubly-extended Reed-Solomon code of codimension 4 as the code associated with the twisted cubic in the projective space.
References
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Book
Projective geometries over finite fields
TL;DR: The first properties of the plane can be found in this article, where the authors define the following properties: 1. Finite fields 2. Projective spaces and algebraic varieties 3. Subspaces 4. Partitions 5. Canonical forms for varieties and polarities 6. The line 7. Ovals 9. Arithmetic of arcs of degree two 10. Cubic curves 12. Arcs of higher degree 13. Blocking sets 14. Small planes 15.
Book
Finite projective spaces of three dimensions
TL;DR: Projective Geometrics Over Finite Fields (OUP, 1979) as mentioned in this paper considers projective spaces of three dimensions over a finite field and examines properties of four and five dimensions, fundamental applications to translation planes, simple groups, and coding theory.
BookDOI
General Galois geometries
TL;DR: In this paper, the authors define Hermitian varieties, Grassmann varieties, Veronese and Segre varieties, and embedded geometries for finite projective spaces of three dimensions.