Projective Plane Of Order Research Articles

Pedal Sets of Unitals in Projective Planes of Order 9 and 16

Pedal Sets of Unitals in Projective Planes of Order 9 and 16

Construction of Qــ Ary(n,Μ,d )ــ Codes in PG (2,9)

This study's main objective is to offer the connection between the projective plane of order nine & the field of coding theory. An incidence matrix has been developed using the parameters n (code length), d (code min. distance), and e (code error correcting). The greatest value of size code M across a limited field of the ninth order has also been determined. There are a few theorems and examples provided.

Enumeration of Sets of Mutually Orthogonal Latin Rectangles

We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural variation of the concept of self-orthogonal Latin squares which is applicable on larger sets of mutually orthogonal Latin squares and MOLR, namely that each Latin rectangle in a set of MOLR is isotopic to each other rectangle in the set. We call such a set of MOLR homogeneous.In the course of doing this, we perform a complete enumeration of non-isotopic sets of $t$ mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$, for all $t < n$. Specifically, we keep track of homogeneous sets of MOLR, as well as sets of MOLR where the autotopism group acts transitively on the rectangles, and we call such sets of MOLR transitive.We build the sets of MOLR row by row, and in this process we also keep track of which of the MOLR are homogeneous and/or transitive in each step of the construction process. We use the prefix stepwise to refer to sets of MOLR with this property.Sets of MOLR are connected to other discrete objects, notably finite geometries and certain regular graphs. Here we observe that all projective planes of order at most 9 except the Hughes plane can be constructed from a stepwise transitive MOLR.

On the (τ^2,τ)-Arcs in PG(2,qq) for τ≤qq

In this article the existence of the arcs in the projective plane of order has been proved. For we have shown that the arcs are complete. The existence of the completes (144,12)-arc in , (256,16)-arc in and - (324,18)arc in have been found.

On partial geometries arising from maximal arcs

The subject of this paper are partial geometries \(pg(s,t,\alpha )\) with parameters \(s=d(d'-1), \ t=d'(d-1), \ \alpha =(d-1)(d'-1)\), \(d, d' \ge 2\). In all known examples, \(q=dd'\) is a power of 2 and the partial geometry arises from a maximal arc of degree d or \(d'\) in a projective plane of order q via a known construction due to Thas [28] and Wallis [34], with a single known exception of a partial geometry pg(4, 6, 3) found by Mathon [22] that is not associated with a maximal arc in the projective plane of order 8. A parallel class of lines is a set of pairwise disjoint lines that covers the point set. Two parallel classes are called orthogonal if they share exactly one line. An upper bound on the maximum number of pairwise orthogonal parallel classes in a partial geometry G with parameters \(pg(d(d'-1),d'(d-1),(d-1)(d'-1))\) is proved, and it is shown that a necessary and sufficient condition for G to arise from a maximal arc of degree d or \(d'\) in a projective plane of order \(q=dd'\) is that both G and its dual geometry contain sets of pairwise orthogonal parallel classes that meet the upper bound. An alternative construction of Mathon’s partial geometry is presented, and the new necessary condition is used to demonstrate why this partial geometry is not associated with any maximal arc in the projective plane of order 8. The partial geometries associated with all known maximal arcs in projective planes of order 16 are classified up to isomorphism, and their parallel classes of lines and the 2-rank of their incidence matrices are computed. Based on these results, some open problems and conjectures are formulated.

Small stopping sets in finite projective planes of order q

A configuration C in a (finite) incidence structure is a subset C of blocks. If every point on a block of C belongs to at least one other block of C, then C is called stopping set (or equivalently full configuration). If smin(q) is the minimal size of a stopping set in a finite projective plane of odd order q, then either smin(q) ≥ q + 5 if $3\not\vert q$ or smin(q) ≥ q + 3 if 3|q. In this note, we prove that smin(q) ≥ q + 5 for any odd q ≠ 3. If q = 3, then smin(3) = 6 and a stopping set of minimal size 6 in PG(2, 3) is the dual set of the symmetric difference of two lines. Also, we study stopping sets of size q + 4 in a finite projective plane of order q.

On t‐designs and s‐resolvable t‐designs from hyperovals

AbstractHyperovals in projective planes turn out to have a link with t‐designs. Motivated by an unpublished work of Lonz and Vanstone, we present a construction for t‐designs and s‐resolvable t‐designs from hyperovals in projective planes of order . We prove that the construction works for . In particular, for the construction yields a family of 5‐ designs. For numerous infinite families of 4‐designs on points with block size can be constructed for any . The construction assumes the existence of a 4‐ design, called the indexing design, including the complete 4‐ design. Moreover, we prove that if the indexing design is s‐resolvable, then so is the constructed design. As a result, many of the constructed designs are s‐resolvable for . We include a short discussion on the simplicity or non‐simplicity of the designs from hyperovals.

Weighted (k,n)-arcs of Type (n−q,n) and Maximum Size of (h,m)-arcs in PG(2,q)

In this paper, we introduce a generalized weighted \((k, n)\)-arc of two types in the projective plane of order \(q\), where \(q\) is an odd prime number. The sided result of this work is finding the largest size of a complete \((h, m)\)-arcs in \(\mathrm{PG}(2, q)\), where \(h\) represents a point of weight zero of a weighted \((k, n)\)-arc. Also, we prove that a \(\big(\frac{q(q-1)}{2}+1, \frac{q+1}{2}\big)\)-arc is a maximal arc in \(\mathrm{PG}(2, q)\).

On the geometric pictures of the non-equivalent smallest images of (k;n)-arcs for n = 2,3 in PG(2,5)

In this paper, the smallest images sets in the PGL(2,5)-orbits of (k;n)-arcs in the projective plane of order 5, for n = 2,3 are classified. Also, this paper presents the incident structure of the orbits for the non-equivalent lexicographically least sets of k-arcs and (k;3)-arcs. Then the geometric pictures for the most interesting group orbits of these images are given.

Some Applications of Coding Theory in the Projective Plane of Order Four

The main aim of this research is to introduce the relationship between the topic of coding theory and the projective plane of order four. The maximum value of size code M over the finite field of order four and an incidence matrix with the parameters, n (length of code), d (minimum distance of code) and e (error-correcting of code) have been constructed. Some examples and theorems have been given.

On some 3-primitive projective planes

We evolve an approach to construction and classification of semifield projective planes with the use of the linear space and spread set. This approach is applied to the problem of existance for a projective plane with the fixed restrictions on collineation group. A projective plane is said to be semifield plane if its coordinatizing set is a semifield, or division ring. It is an algebraic structure with two binary operation which satisfies all the axioms for a skewfield except (possibly) associativity of multiplication. A collineation of a projective plane of order p 2n ( p > 2 be prime) is called Baer collineation if it fixes a subplane of order p n pointwise. If the order of a Baer collineation divides p n − 1 but does not divide p i − 1 for i < n then such a collineation is called p -primitive. A semifield plane that admit such collineation is a p -primitive plane. M. Cordero in 1997 construct 4 examples of 3-primitive semifield planes of order 81 with the nucleus of order 9, using a spread set formed by 2 × 2-matrices. In the paper we consider the general case of 3-primitive semifield plane of order 81 with the nucleus of order ≤ 9 and a spread set in the ring of 4 × 4-matrices. We use the earlier theoretical results obtained independently to construct the matrix representation of the spread set and autotopism group. We determine 8 isomorphism classes of 3-primitive semifield planes of order 81 including M. Cordero examples. We obtain the algorithm to optimize the identification of pair-isomorphic semifield planes, and computer realization of this algorithm. It is proved that full collineation group of any semifield plane of order 81 is solvable, the orders of all autotopisms are calculated. We describe the structure of 8 non-isotopic semifields of order 81 that coordinatize 8 nonisomorphic 3-primitive semifield planes of order 81. The spectra of its multiplicative loops of non-zero elements are calculated, the left-, right-ordered spectra, the maximal subfields and automorphisms are found. The results obtained illustrate G. Wene hypothesis on left or right primitivity for any finite semifield and demonstrate some anomalous properties. The methods and algorithsm demonstrated can be used for construction and investigation of semifield planes of odd order p n for p ≥ 3 and n ≥ 4.

Triples of orthogonal Latin and Youden rectangles for small orders

AbstractWe have performed a complete enumeration of nonisotopic triples of mutually orthogonal Latin rectangles for . Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this, we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of rectangles which are formed by extending mutually orthogonal triples with nontrivial autotopisms one row at a time, and requiring that the autotopism group is nontrivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of rectangles, formed by juxtaposing two selected copies of complete sets of mutually orthogonal Latin squares of order 4.

Some application of coding theory in the projective plane of order three

The main aim of this paper is to introduce the relationship between the topic of coding theory and the projective plane of order three. The maximum value of size of code over finite field of order three and an incidence matrix with the parameters, (length of code), (minimum distance of code) and (error-correcting of code ) have been constructed. Some examples and theorems have been given.

A family of extremal hypergraphs for Ryser's conjecture

Ryser's Conjecture states that for any r-partite r-uniform hypergraph, the vertex cover number is at most r−1 times the matching number. This conjecture is only known to be true for r≤3 in general and for r≤5 if the hypergraph is intersecting. There has also been considerable effort made for finding hypergraphs that are extremal for Ryser's Conjecture, i.e. r-partite hypergraphs whose cover number is r−1 times its matching number. Aside from a few sporadic examples, the set of uniformities r for which Ryser's Conjecture is known to be tight is limited to those integers for which a projective plane of order r−1 exists.We produce a new infinite family of r-uniform hypergraphs extremal to Ryser's Conjecture, which exists whenever a projective plane of order r−2 exists. Our construction is flexible enough to produce a large number of non-isomorphic extremal hypergraphs. In particular, we define what we call the Ryser poset of extremal intersecting r-partite r-uniform hypergraphs and show that the number of maximal and minimal elements is exponential in r.This provides further evidence for the difficulty of Ryser's Conjecture.

One-factorisations of complete graphs arising from ovals in finite planes

In this paper we use the geometry of finite planes to set up a procedure for the construction of one-factorisations of the complete graph. Let π be a projective plane of order n−1 with n even containing an oval Ω, and regard Ω as the vertex set of the complete graph Kn. Then any one-factorisation of Kn has a representation by a partition of the external points to Ω whose components are of size n2 and meet every tangent to Ω in a unique point. Our goal is to construct such partitions from nice geometric configurations in the Desarguesian plane of order q with q=ph and p>2 prime.

On the Pseudoachromatic Index of the Complete Graph III

Let $ \Pi_q $ be the projective plane of order $ q $, let $\psi(m):=\psi(L(K_m))$ the pseudoachromatic number of the complete line graph of order $ m $, let $ a\in \{ 3,4,\dots,\tfrac{q}{2}+1 \} $ and $ m_a=(q+1)^2-a $. In this paper, we improve the upper bound of $ \psi(m) $ given by Araujo-Pardo et al. [J Graph Theory 66 (2011), 89--97] and Jamison [Discrete Math. 74 (1989), 99--115] in the following values: if $ x\geq 2 $ is an integer and $m\in \{4x^2-x,\dots,4x^2+3x-3\}$ then $\psi(m) \leq 2x(m-x-1)$. On the other hand, if $ q $ is even and there exists $ \Pi_q $ we give a complete edge-colouring of $ K_{m_a} $ with $(m_a-a)q$ colours. Moreover, using this colouring we extend the previous results for $a=\{-1,0,1,2\}$ given by Araujo-Pardo et al. in [J Graph Theory 66 (2011), 89--97] and [Bol. Soc. Mat. Mex. (2014) 20:17--28] proving that $\psi(m_a)=(m_a-a)q$ for $ a\in \{3,4,\dots,\left\lceil \frac{1+\sqrt{4q+9}}{2}\right\rceil -1 \} $.

Segment Combining and Patching for Short Packet Forward Error Control Coding

Modern networks are popular with a large number of short packets generated by many light weight, low power devices. We propose a new approach, segment combining and patching (SCP), for forward error control coding suitable for short packets, where the encoder logically divides a full packet into a number of partially overlapped subpackets based on a Finite Projective Plane of order m, and transmits the full packet n times. Upon receiving all n replicas of a full packet, the decoder generates an additional combined packet using Maximum Ratio Combining. By selecting non-faulty subpackets through Cyclic Redundancy Checks, both inter-cluster and intra-cluster patching are then performed on the n + 1 clusters of subpackets until no new non-faulty subpackets are generated. Simulation results in both AWGN and block fading channels demonstrate that the SCP is an effective approach for forward error control coding for short packet communication in embedded networks, sensor networks, and so on. Lastly, we provide a theory of SCP to show how it works.

Partitioning 3‐arcs into Steiner Triple Systems

AbstractIn this article, it is shown that there is a partitioning of the set of 3‐arcs in a projective plane of order three into nine pairwise disjoint Steiner triple systems of order 13.

On resolvable Steiner 2-designs and maximal arcs in projective planes

A combinatorial characterization of resolvable Steiner 2-(v, k, 1) designs embeddable as maximal arcs in a projective plane of order $$(v-k)/(k-1)$$(v-k)/(k-1) is proved, and a generalization of a conjecture by Andries Brouwer (Geometries and groups, Springer, Heidelberg, 1981) is formulated.

Dominating Sets in Projective Planes

AbstractWe describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result that shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a dominating set in a projective plane of order is smaller than (i.e., twice the size of a Baer subplane), then it contains either all but possibly one points of a line or all but possibly one lines through a point. Furthermore, we completely characterize dominating sets of size at most . In Desarguesian planes, we could rely on strong stability results on blocking sets to show that if a dominating set is sufficiently smaller than 3q, then it consists of the union of a blocking set and a covering set apart from a few points and lines.