Subspaces Of PG Research Articles

Partitions by subgeometries of projective plane over a galois field of order 3n, n = 1, 2, 3

The purpose of this thesis is to split the projective plane of PG (2,9) into seven disjoint projective subplanes PG (2,3) and thirteen disjoint complete arcs of degree two and size seven. The projective line of order twenty-seven PG (1,27) has been partitioned into seven disjoint projective sublines PG (1,3) and the number of inequivalent -sets which are unordered sets of four points is classified. The group action on projective lines PG (1,3n) and projective PG (1,3n), n = 1, 2, 3 planes is explained and we have introduced theorems and examples and the subspaces of PG (1,3n) and PG (2,3n), n = 1, 2, 3 are shown. Each of these partitions gives rise to an error-correcting code that corrects the maximum possible number of errors for its length. GAP-Groups–Algorithms, Programming a system for computational discrete algebra has been applied.

Graph decompositions in projective geometries

AbstractLet PG be the ‐dimensional projective space over and let be a simple graph of order for some . A design over is a collection of graphs (blocks) isomorphic to with the following properties: the vertex set of every block is a subspace of PG; every two distinct points of PG are adjacent in exactly blocks. This new definition covers, in particular, the well‐known concept of a design over corresponding to the case that is complete. In this study of a foundational nature we illustrate how difference methods allow us to get concrete nontrivial examples of ‐decompositions over or for which is a cycle, a path, a prism, a generalized Petersen graph, or a Moebius ladder. In particular, we will discuss in detail the special and very hard case that is complete and , that is, the Steiner 2‐designs over a finite field. Also, we briefly touch the new topic of near resolvable designs over . This study has led us to some (probably new) collateral problems concerning difference sets. Supported by multiple examples, we conjecture the existence of infinite families of Γ‐decompositions over a finite field that can be obtained by suitably labeling the vertices of Γ with the elements of a Singer difference set.

On the algebraic variety [formula omitted

The variety V r , t is the image under the Grassmannian map of the ( t − 1 ) -subspaces of PG ( r t − 1 , q ) of the elements of a Desarguesian spread. We investigate some properties of this variety, with particular attention to the case r = 2 : in this case we prove that every t + 1 points of the variety are in general position and we give a new interpretation of linear sets of PG ( 1 , q t ) .

2-Spreads and Transitive and Orthogonal 2-Parallelisms of PG(5, 2)

A 2-spread is a set of two-dimensional subspaces of PG(d, q), which partition the point set. We establish that up to equivalence there exists only one 2-spread of PG(5, 2). The order of the automorphism group preserving it is 10584. A 2-parallelism is a partition of the set of two-dimensional subspaces by 2-spreads. There is a one-to-one correspondence between the 2-parallelisms of PG(5, 2) and the resolutions of the 2-(63,7,15) design of the points and two-dimensional subspaces. Sarmiento (Graphs and Combinatorics 18(3):621–632, 2002) has classified 2-parallelisms of PG(5, 2), which are invariant under a point transitive cyclic group of order 63. We classify 2-parallelisms with automorphisms of order 31. Among them there are 92 2-parallelisms with full automorphism group of order 155, which is transitive on their 2-spreads. Johnson and Montinaro (Results Math 52(1–2):75–89, 2008) point out that no transitive t-parallelisms of PG(d, q) have been constructed for t > 1. The 92 transitive 2-parallelisms of PG(5, 2) are then the first known examples. We also check them for mutual orthogonality and present a set of ten mutually orthogonal resolutions of the geometric 2-(63,7,15) design.

Blocking sets in PG(2, qn) from cones of PG(2n, q)

Let ? and $${\bar B}$$ be a subset of ? = PG(2n?1,q) and a subset of PG(2n,q) respectively, with ? ? PG(2n,q) and $${{\bar B}\not\subset \Sigma}$$ . Denote by K the cone of vertex ? and base $${\bar B}$$ and consider the point set B defined by $$B=\big(K{\setminus}\Sigma\big) \cup \{X\in \S\, : \, X\cap K\neq \emptyset\},$$ in the Andre, Bruck-Bose representation of PG(2,qn) in PG(2n,q) associated to a regular spread $${\cal S}$$ of PG(2n?1,q). We are interested in finding conditions on $${\bar B}$$ and ? in order to force the set B to be a minimal blocking set in PG(2,qn) . Our interest is motivated by the following observation. Assume a Property ? of the pair (?, $${\bar B}$$ ) forces B to turn out a minimal blocking set. Then one can try to find new classes of minimal blocking sets working with the list of all known pairs (?, $${\bar B}$$ ) with Property ?. With this in mind, we deal with the problem in the case ? is a subspace of PG(2n?1,q) and $${\bar B}$$ a blocking set in a subspace of PG(2n,q); both in a mutually suitable position. We achieve, in this way, new classes and new sizes of minimal blocking sets in PG(2,qn), generalizing the main constructions of [14]. For example, for q = 3h, we get large blocking sets of size qn + 2 + 1 (n? 5) and of size greater than qn+2 + qn?6 (n? 6). As an application, a characterization of Buekenhout-Metz unitals in PG(2,q2k) is also given.

The invariant factors of the incidence matrices of points and subspaces in 𝑃𝐺(𝑛,𝑞) and 𝐴𝐺(𝑛,𝑞)

We determine the Smith normal forms of the incidence matrices of points and projective ( r − 1 ) (r-1) -dimensional subspaces of PG ⁡ ( n , q ) \operatorname {PG}(n,q) and of the incidence matrices of points and r r -dimensional affine subspaces of AG ⁡ ( n , q ) \operatorname {AG}(n,q) for all n n , r r , and arbitrary prime power q q .

Blocking Subspaces By Lines In PG(n, q)

This paper studies the cardinality of a smallest set $${\user1{T}}$$ of t-subspaces of the finite projective spaces PG(n, q) such that every s-subspace is incident with at least one element of $${\user1{T}}$$, where 0 ≤ t < s ≤ n. This is a very difficult problem and the solution is known only for very few families of triples (s, t, n). When the answer is known, the corresponding blocking configurations usually are partitions of a subspace of PG(n, q) by subspaces of dimension t. One of the exceptions is the solution in the case t = 1 and n = 2s. In this paper, we solve the case when t = 1 and 2s < n ≤ 3s-3 and q is sufficiently large.

On the Orthogonality of Geometric Codes

We solve some problems concerning the orthogonality of geometric codes associated with sets of i- and j-dimensional subspaces of PG(n, q). Various applications are found, and we discuss all the interesting cases in small dimensional spaces.

Critical Exponents, Colines, and Projective Geometries

In [9, p. 469], Oxley made the following conjecture, which is a geometric analogue of a conjecture of Lovász (see [1, p. 290]) about complete graphs.Conjecture 1.1.Let G be a rank-n GF(q)-representable simple matroid with critical exponent n − γ. If, for every coline X in G, c(G/X; q) = c(G; q) − 2 = n − γ − 2, then G is the projective geometry PG(n − 1, q).We shall call the rank n, the critical ‘co-exponent’ γ, and the order q of the field the parameters of Oxley's conjecture. We exhibit several counterexamples to this conjecture. These examples show that, for a given prime power q and a given positive integer γ, Oxley's conjecture holds for only finitely many ranks n. We shall assume familiarity with matroid theory and, in particular, the theory of critical problems. See [6] and [9].A subset C of points of PG(n − 1, q) is a (γ, k)-cordon if, for every k-codimensional subspace X in PG(n − 1, q), the intersection C ∩ X contains a γ-dimensional subspace of PG(n − 1, q). In this paper, our primary interest will be in constructing (γ, 2)-cordons. With straightforward modifications, our methods will also yield (γ, k)-cordons.Complements of counterexamples to Oxley's conjecture are (γ, 2)-cordons.

Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 1

The flag geometry Γ=(P, L, I) of a finite projective plane Π of order s is the generalized hexagon of order (s, 1) obtained from Π by putting P equal to the set of all flags of Π, by putting L equal to the set of all points and lines of Π, and where I is the natural incidence relation (inverse containment), i.e., Γ is the dual of the double of Π in the sense of H. Van Maldeghem (1998, “Generalized Polygons,” Birkhäuser Verlag, Basel). Then we say that Γ is fully and weakly embedded in the finite projective space PG(d, q) if Γ is a subgeometry of the natural point-line geometry associated with PG(d, q), if s=q, if the set of points of Γ generates PG(d, q), and if the set of points of Γ not opposite any given point of Γ does not generate PG(d, q). In an earlier paper, we have shown that the dimension d of the projective space belongs to {6, 7, 8}, and that the projective plane Π is Desarguesian. Furthermore, we have given examples for d=6, 7. In the present paper we show that for d=6, only these examples exist, and we also partly handle the case d=7. More precisely, we completely classify the full and weak embeddings of Γ (Γ as above) in the case that there are two opposite lines L, M of Γ with the property that the subspace of PG(d, q) generated by all lines of Γ meeting either L or M has dimension 6 (which is the case for all embeddings in PG(d, q), d∈{6, 7}). Together with Parts 2 and 3, this will provide the complete classification of all full and weak embeddings of Γ.

A Family of d -dimensional Dual Hyperovals in P G(2 d + 1, 2)

With the affine part of an oval we associate a family of d -subspaces ofPG (2 d+ 1, 2) which can be thought of as a higher dimensional analogue of a hyperoval. The isomorphisms among such families together with their automorphisms are determined when they come from translation ovals.

On the classification of geometric codes by polynomial functions

It is shown how to represent algebraically all functions that have a zero sum on all μ-dimensional subspaces ofPG(n,q) or ofAG(n,q). In this way one can calculate the dimensions of related codes, or one can represent interesting sets of points by functions.

Embedding pseudo-complements of quadrics in PG( n, q), n⩾3, q⩾2

Let S be a finite, planar, linear space of dimension n⩾3 such that (1) each line has q − 1, q, or q + 1 points; (2) in any subspace R, the number of lines on any point of R is (q dimR − 1) (q − 1) , where q⩾2. We prove that S embeds in a unique way in PG( n, q). If in addition S has at most q n points, it follows, using a result of Tallini, that S is the complement in PG( n, q) of a parabolic or hyperbolic quadric, a parabolic quadric plus a subspace of its nucleus space, a cone projecting from a PG( n − 3, q) a plane ( q + 1)-arc plus a subspace of the PG( n − 2, q) joining the knot of the arc with the PG( n − 3, q), or a hyperplane along with a subspace of PG( n, q).

On the graphical characterization of the projective space over a finite field

We shall construct the graphs whose vertex set is the set of all k-dimension projective subspaces of PG(n, q), two vertices being joined by an edge whenever they have a (k 1 )-dimension projective subspace in common. Then these graphs satisfy the condition (ii), and for two vertices c1 and B, with d(cr, /I) = 2, d(cr)nd(/?) is isomorphic to L,(t). (d is a distance function.) I received the advice that Theorem A is generalized by applying the main theorem of Cohen [4] as follows: 143 0095-8956/85 $3.00

Characterizations of Designs Constructed from Affine and Projective Spaces

The design P n -1 ( n , q ) of points and hyperplanes of the projective space PG ( n , q ) may be considered as a “completion” of the design A n -1 ( n , q ) of points and hyperplanes of the affine space AG ( n , q ). We obtain some characterizations of P n -1 ( n , q ) as a “completion” of A n -1 ( n , q ). Moreover, we give direct characterizations of the designs P t ( n , q ) and A t ( n , q ) of points and t -dimensional subspaces of PG ( n , q ) and AG ( n , q ) respectively.

On the transitivity of the orthogonal and symplectic groups in projective space

1.Introduction. Let Ω be a non-singular quadric in ann-dimensional projective spacePG(n, K) whose coordinate field isK. With respect to Ω the linear subspaces ofPG(n, K) fall into various types: the subspaces of a given type each have the same dimension and the same geometrical kind of quadric section with Ω. Each element of the collineation group Γ preserving Ω takes a subspace into one of the same type, but Γ may divide the subspaces of a given type into several transitivity classes or orbits.