Plane PG Research Articles

Classifying Sets of Type (4,n) in PG(3,q)

In the present work, we classify sets of type (4,n) in PG(3,q). We prove that PG(3,q), apart from the planes of PG(3,3), contains only sets of type (4,n) with standard parameters. Thus, somewhat surprisingly, we conclude that there are no sets of type (4,n) in PG(3,q), q ≠ 3, with non-standard parameters.

Large (k,3)-arcs in PG(2,19) and the related linear codes

A (n,r)-arc in a projective plane PG(2,q) is a set of n points such that some r, but no r+1 of them, are collinear. A (n,r)-arc is called complete if it is not contain in a (n+1,r)-arc. A linear -code over a finite field is a k-dimensional subspace of with minimum hamming distance d and length n. A code with parameters with Griesmer bound , is called Griesmer code. The major aim of this research is to find large size for the complete (k,3)-arcs in the projective plane of order nineteen PG(2,19) using the method of secants distributions, and the disjoint union of arcs, as well as, adding and removing points to (from) particular conic respectively. Also, we find the Griesmer codes that correspond to each large complete (k,3)-arcs, k=29,30,31. We introduced 20 inequivalent (29,3)-arcs up to secant distribution, 10 of them are complete. Also, we introduced 8 inequivalent (30,3)-arcs up to secant distribution, 2 of them are complete. Moreover, we construct 3 inequivalent complete (31,3)-arcs up to secant distribution, and then find the corresponding linear codes to some of the (29,3)-arcs, (30,3)-arcs and (31,3)-arcs. In particular, we established 3 types of Griesmer codes, and we find the weight enumerator that correspond to each one of them.

Finding the complete List of Different K-Brackets for the Projective Plane PG (2, 8)

A k-arc in a plane PG (2, q) is a set of k point such that every line in the plane intersect it in at most two points and there is a line intersect it in exactly two points. A k-arc is complete if there is no k+1-arc containing it. This thesis is concerned with studies a k-arcs, k=4, 5,…., 10 and classification of protectively distinct k-arcs and distinct arcs under collineation. We prove by using computer program that the only complete k-arcs is for, k= 6, 10.

A higgledy-piggledy set of planes based on the ABB-representation of linear sets

In this paper, we investigate the André/Bruck-Bose representation of certain Fq-linear sets contained in a line of PG(2,qt). We show that scatteredFq-linear sets of rank 3 in PG(1,q3) correspond to particular hyperbolic quadrics and that Fq-linear clubs in PG(1,qt) are linked to subspaces of a certain 2-design based on normal rational curves; this design extends the notion of a circumscribed bundle of conics. Finally, we use these results to construct optimal higgledy-piggledy sets of planes in PG(5,q).

On the geometry of a (q + 1)-arc of PG(3,q), q even

In PG(3,q), q=2n, n≥3, let Ah,q={(1,t,t2h,t2h+1)|t∈Fq}∪{(0,0,0,1)}, with gcd⁡(n,h)=1, be a (q+1)-arc and let Gh≃PGL(2,q) be the stabilizer of Ah,q in PGL(4,q). The Gh-orbits on points, lines and planes of PG(3,q), together with the point-plane incidence matrix with respect to the Gh-orbits on points and planes of PG(3,q) are determined. The point-line incidence matrix with respect to the G1-orbits on points and lines of PG(3,q) is also considered. In particular, for a line ℓ belonging to a given line G1-orbit, say L, the point G1-orbit distribution of ℓ is either explicitly computed or it is shown to depend on the number of elements x in Fq (or in a subset of Fq) such that Trq|2(g(x))=0, where g is an Fq-map determined by L.

Construction of q-ary(n,M,d)-codes in PG(2,16)

The goal of this paper was to study the applications of the projective plane PG (2, q) over a Galois field of order q in the projective q-ary (n, M, d) -code such that the parameters length of code n, the maximum value size code M, and the minimum distance d with the error-correcting e according to an incidence matrix have been calculated. Also, this research provides examples and theorems of links between the combinatorial structures and coding theory. The method of the research depends on the classification of the points and lines in PG (2, q).

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.

Super-regular Steiner 2-designs

A design is additive under an abelian group G (briefly, G-additive) if, up to isomorphism, its point set is contained in G and the elements of each block sum up to zero. The only known Steiner 2-designs that are G-additive for some G have block size which is either a prime power or a prime power plus one. Indeed they are the point-line designs of the affine spaces AG(n,q), the point-line designs of the projective planes PG(2,q), the point-line designs of the projective spaces PG(n,2) and a sporadic example of a 2-(8191,7,1) design. In the attempt to find new examples, possibly with a block size which is neither a prime power nor a prime power plus one, we look for Steiner 2-designs which are strictly G-additive (the point set is exactly G) and G-regular (any translate of any block is a block as well) at the same time. These designs will be called “G-super-regular”. Our main result is that there are infinitely many values of v for which there exists a super-regular, and therefore additive, 2-(v,k,1) design whenever k is neither singly even nor of the form 2n3≥12. The case k≡2 (mod 4) is a genuine exception whereas k=2n3≥12 is at the moment a possible exception. We also find super-regular 2-(pn,p,1) designs with p∈{5,7} and n≥3 which are not isomorphic to the point-line design of AG(n,p).

Simplicial complexes from finite projective planes and colored configurations

In the 7-vertex triangulation of the torus, the 14 triangles can be partitioned as T1⊔T2, such that each Ti represents the lines of a copy of the Fano plane PG(2,F2). We generalize this observation by constructing, for each prime power q, a simplicial complex Xq with q2+q+1 vertices and 2(q2+q+1) facets consisting of two copies of PG(2,Fq). Our construction works for any colored k-configuration, defined as a k-configuration whose associated bipartite graph G is connected and has a k-edge coloring χ:E(G)→[k], such that for all v∈V(G), a,b,c∈[k], following edges of colors a,b,c,a,b,c from v brings us back to v. We give one-to-one correspondences between (1) Sidon sets of order 2 and size k+1 in groups with order n, (2) linear codes with radius 1 and index n in the lattice Ak, and (3) colored (k+1)-configurations with n points and n lines. (The correspondence between (1) and (2) is known.) As a result, we suggest possible topological obstructions to the existence of Sidon sets, and in particular, planar difference sets.

Splitting of PG(1,27) by Sets, Orbits, and Arcs on the Conic

This research aims to give a splitting structure of the projective line over the finite field of order twenty-seven that can be found depending on the factors of the line order. Also, the line was partitioned by orbits using the companion matrix. Finally, we showed the number of projectively inequivalent -arcs on the conic through the standard frame of the plane PG(1,27)

Characterising the secant lines of Q(4,q), q even

We show that a set A of lines in PG(4,q), q even, is the set of secant lines of a parabolic (non-singular) quadric if and only if A satisfies the following three conditions:(I)every point of PG(4,q) lies on 0,12q3 or q3 lines of A;(II)every plane of PG(4,q) contains 0, 12q(q+1) or q2 lines of A; and(III)every hyperplane of PG(4,q) contains 12q2(q2+1), 12q3(q+1) or 12q2(q+1)2 lines of A.

Study subsets in Unite projective planes and their applications to Error-Correcting codes

The main objective of this paper is to classify the special sets in the projective planes PG (2, t) with the values of t= 17,31,37, and we found the arcs of the second degree (k,2) - arcs in PG (2, t) or third-degree (k,3)-arcs in PG(2, t), and the parameters of codes n, k, d was calculated and whether they are complete or incomplete, and the corresponding codes were formed.

Special Arcs in Projective Plane Over Galois Field of Order Sixteen

The main objective of this paper is to classify the special sets in the projective planes PG(2, t) with the value of t = 16, and we found the arcs of the third degree (k, 3) arcs in P G(2, t), and the parameters of cods n, k, d were calculated and whether they are complete or incomplete, and the corresponding codes were formed.

On the non-trivial minimal blocking sets in binary projective spaces

We prove that a non-trivial minimal blocking set with respect to hyperplanes in PG(r,2), r≥3, is a skeleton contained in some s-flat with odd s≥3. We also consider non-trivial minimal blocking sets with respect to lines and planes in PG(r,2), r≥3. Especially, we show that there are exactly two non-trivial minimal blocking sets with respect to lines and six non-trivial minimal blocking sets with respect to planes up to projective equivalence in PG(4,2). A characterization of an elliptic quadric in PG(5,2) as a special non-trivial minimal blocking set with respect to planes meeting every hyperplane in a non-trivial minimal blocking sets with respect to planes is also given.

On the balanced upper chromatic number of finite projective planes

In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph H, the maximum number k for which there is a balanced rainbow-free k-coloring of H is called the balanced upper chromatic number of the hypergraph. We confirm the conjecture of Araujo-Pardo, Kiss and Montejano by determining the balanced upper chromatic number of the desarguesian projective plane PG(2,q) for all q. In addition, we determine asymptotically the balanced upper chromatic number of several families of non-desarguesian projective planes and also provide a general lower bound for arbitrary projective planes using probabilistic methods which determines the parameter up to a multiplicative constant.

The construction for the arcs (8,4)-from the two arcs (7,4)-in PG (2,q), q=5

In the current research, the points and lines of the projective plane of order five were constructed, this was followed by examining the arcs designated (k,n)-in this plane. As a result it is concluded that there arcs do exist having set of arcs (8,4)-where the number 8 represented the number of points of the arc while the number 4 represented the largest number of points lie on one line exactly which we obtain there arcs from arcs (7,4)-by using special method we reached it where we study the general properties of (k,n)-arcs in the projective plane PG(2,q).

A characterization of the family of secant lines to a hyperbolic quadric in [formula omitted], [formula omitted] odd

We give a combinatorial characterization of the family of lines of PG(3,q), q≥7 odd, which meet a hyperbolic quadric in two points (the so called secant lines) using their intersection properties with the points and planes of PG(3,q).

The group action on the finite projective planes of orders 53, 61, 64

The goal of this research is to partition the projective plane PG(2, q) over Galois Field of order q where q = 53, 61, 64. Classify the group action cyclic groups on PG(2, q), when the number of points and lines is equal to q 2 + q + 1 which is not a prime have been described. Study the properties of the subsets (orbits, points, lines, planes, triangles) geometrically and algebraically. Also establish the projective codes and arcs with different sizes and degrees by using the program of GAP.

Study of orbits on the finite projective plane

The aims of this paper are to classify the group action on a projective plane PG(2, q), when q 2 + q + 1 is not a prime, and then describe the geometry of these orbits with values of q = 43, 47, 49. Investigate partitions of PG(2, q) with q = 49, into subgeometries PG(2, q) when q = 7. Also establish codes and arcs with different degrees.

Application of Algebraic Geometry In Three Dimensional projective space PG (3,7)

The main goal of this work is to construct surfaces and complete arcs in the projective 3 – space PG (3, q) over Galois fields GF (p), p=7. Which represents applications of algebraic geometry in three-dimensional projective space PG (3, P), where p=7 which is a (k, ƪ)-span. We get the following results. First, we found the points, lines and planes in PG (3,7) and we construct (k, ƪ)-span which is a set of k lines no two of which intersect. We prove that the maximum complete (k, ƪ)-span in PG (3,7) is (50, ƪ)-span, which is the equal to all the points of the space that is called a spread. Second in general we prove geometrical rule the total number of Spread in projective space PG (3, p) where p is prime, P ≥ 2 is p 2 + l.