Laplacian Integral of Particular Steiner System

The notion of a hypergraph is motivated by a graph. In graph, every edge contains of two vertices. However, a hypergraph edges contains more than two vertices. In this article use hyperedge to mention edge of hypergraph. A finite projective plane of order n, denoted by , is a linear intersecting hypergraph. In this research finite projective plane order is Laplacian integral.

On the Embedding of an Arc into Cubic Curves in a Finite Projective Plane of Order Seven

The main aims of this research are to find the stabilizer groups of cubic curves over a finite field of order 7 and studying the properties of their groups and then constructing the arcs of degree 2 which are embedding in cubic curves of even size which are considering as the arcs of degree 3. Also drawing all these arcs.

Classification of the Hyperovals in PG(2,64)

In this paper, we present a full classification of the hyperovals in the finite projective plane $\mathrm{PG}(2,64)$, showing that there are exactly 4 isomorphism classes. The techniques developed to obtain this result can be applied more generally to classify point sets with $0$ or $2$ points on every line, in a broad range of highly symmetric incidence structures.

Cubic Arcs in the Projective Plane Over a Finite Field of Order Twenty Three

In this research we are interested in finding all the different cubic curves over a finite projective plane of order twenty-three, learning which of them is complete or not, constructing the stabilizer groups of the cubics in, studying the properties of these groups, and, finally, introducing the relation between the subject of coding theory and the projective plane of order twenty three.

Minimal Multiple Blocking Sets

We prove that a minimal $t$-fold blocking set in a finite projective plane of order $n$ has cardinality at most \[\frac{1}{2} n\sqrt{4tn - (3t + 1)(t - 1)} + \frac{1}{2} (t - 1)n + t.\] This is the first general upper bound on the size of minimal $t$-fold blocking sets in finite projective planes and it generalizes the classical result of Bruen and Thas on minimal blocking sets. From the proof it directly follows that if equality occurs in this bound then every line intersects the blocking set $S$ in either $t$ points or $\frac{1}{2}(\sqrt{4tn - (3t + 1)(t - 1)} + t - 1) + 1$ points. We use this to show that for $n$ a prime power, equality can occur in our bound in exactly one of the following three cases: (a) $t = 1$, $n$ is a square and $S$ is a unital; (b) $t = n - \sqrt{n}$, $n$ is a square and $S$ is the complement of a Baer subplane; (c) $t = n$ and $S$ is equal to the set of all points except one. For a square prime power $q$ and $t \leq \sqrt{q} + 1$, we give a construction of a minimal $t$-fold blocking set $S$ in $\mathrm{PG}(2,q)$ with $|S| = q\sqrt{q} + 1 + (t - 1)(q - \sqrt{q} + 1)$. Furthermore, we obtain an upper bound on the size of minimal blocking sets in symmetric $2$-designs and use it to give new proofs of other known results regarding tangency sets in higher dimensional finite projective spaces. We also discuss further generalizations of our bound. In our proofs we use an incidence bound on combinatorial designs which follows from applying the expander mixing lemma to the incidence graph of these designs.

Classification of Arcs in Finite Projective Plane of Order Sixteen

The aim of the paper is to classify certain geometric structures, called arcs. The main computing tool is the mathematical programming language GAP. In the plane PG(2,16),the important arcs are called complete and are those that cannot be increased to a larger arc. So far, all arcs up to size eighteen have been classified. Each of these arcs gives rise to an error-correcting code that corrects the maximum possible number of errors for its length.