Configurations in finite projective planes

The main purpose of this paper is to show how partial recursive functions and isols can be used to generalize the following three well-known theorems of combinatorial theory.(I) For every finite projective plane Π there is a unique number n such that Π has exactly n2 + n + 1 points and exactly n2 + n + 1 lines.(II) Every finite projective plane of order n can be coordinatized by a finite planar ternary ring of order n. Conversely, every finite planar ternary ring of order n coordinatizes a finite projective plane of order n.(III) There exists a finite projective plane of order n if and only if there exist n − 1 mutually orthogonal Latin squares of order n.

Let Π be a finite projective plane of order n , and let G be a large abelian (or, more generally, quasiregular) collineation group of Π; to be specific, we assume ∣ G ∣ ( n 2 + n + 1)/2. Such planes have been classified into eight cases by Dembowski and Piper in 1967. We survey the present state of knowledge about the existence and structure of such planes. We also discuss some geometric applications, in particular to the construction of arcs and ovals. Technically, a recurrent theme will be the amazing strength of the approach using various types of difference sets and the machinery of integral group rings. Introduction A projective plane is a geometry consisting of points and lines such that any two distinct lines meet in exactly one point, any two distinct points are on exactly one common line, and there are four points no three of which are collinear. Well-known examples are provided by the desarguesian (or classical ) projective planes PG(2, q ), q a prime power, which can be defined over the finite field GF( q ) as follows: points and lines are the 1- and 2-dimensional subspaces of the vector space GF( q ) 3 , and a point is incident with a line if and only if it is a subset of the line. A standard reference for arbitrary projective planes is Hughes and Piper [104]; for the desarguesian case, see Hirschfeld [89].

The matching number \(\alpha '(H)\) of a hypergraph H is the size of a largest matching in H, where a matching is a set of pairwise disjoint edges in H. A dominating set in H is a subset D of vertices of H such that for every \(v\in V(H)\setminus D\) there exists \(u\in D\) such that u and v lie in an edge of H, and the domination number of H, denoted by \(\gamma (H)\), is the minimum cardinality of a dominating set in H. It was shown that a Ryser-like inequality \(\gamma (H)\le (r-1)\alpha '(H)\) holds for hypergraphs H of rank r. In particular, for intersecting hypergraphs H of rank r, \(\gamma (H)\le r-1\), since \(\alpha '(H)=1\). The linear intersecting hypergraphs of rank \(2\le r\le 4\) achieving the equality \(\gamma (H)=r-1\) have been characterized. In this paper we show that all the 5-uniform linear intersecting hypergraphs H with equality \(\gamma (H)=r-1\) are generated by the finite projective plane of order three.

We apply an improvement of the Delsarte LP-bound to give a new proof of the non-existence of finite projective planes of order 6, and uniqueness of finite projective planes of order 7. The proof is computer aided, and it is also feasible to apply to higher orders like 8, 9 and, with further improvements, possibly 10 and 12.

We present a fairly elementary, self-contained proof of the nonexistence of a finite projective plane of order $6$. Our approach is motivated by theory of binary codes but does not appeal to it directly.

We study a construction of the Mathieu group M 12 using a game reminiscent of Loyd's "15-puzzle." The elements of M 12 are realized as permutations on 12 of the 13 points of the finite projective plane of order 3. There is a natural extension to a "pseudogroup" M 13 acting on all 13 points, which exhibits a limited form of sextuple transitivity. Another corollary of the construction is a metric, akin to that induced by a Cayley graph, on both M 12 and M 13. We develop these results, and extend them to the double covers and automorphism groups of M 12 and M 13, using the ternary Golay code and 12 × 12 Hadamard matrices. In addition, we use experimental data on the quasi-Cayley metric to gain some insight into the structure of these groups and pseudogroups.

Due to the large number of light weight, low power devices in the future Internet, an efficient distributed multiple access protocol that can support a small fixed number of machines with continuous traffic along with a large variable number of devices with sporadic traffic is needed. The traffic in the future Internet has fundamentally different requirements: first, the accessing devices are of low power, which limit the transmission range; secondly, the delay is very stringent; thirdly, the number of active devices is time varying. In this paper, we propose a finite projective plane (FPP) based multiple access protocol (FISH) to meet such requirements. A low power node can transmit the replicas of a data packet in multiple time slots to achieve the same range of transmission as a high power node at one time. The initialization time slot of a frame allows for the estimation of the number of active nodes and the adjustment of the size of the frame to maximize the throughput. The interference cancellation and conflict resolution scheme of the FISH protocol is based on the theory of FPP. By using the property that the union of the sets of the FPP sharing one common point is a complete set, we are able to extract each of the individual user signals. One distinguishing feature of the FISH protocol is the amplification of the signal-to-noise ratio (SNR), which can either extend the transmission range or improve the bit error rate without sacrificing the throughput. In the performance evaluation, we demonstrate that the throughput is identical to the Slotted Aloha, while the SNR is magnified m times, where m is the order of the FPP. The proposed FISH protocol is of great use in the machine-to-machine communications as well as traditional cellular and satellite systems.

ABSTRACTLet be a prime power and be a natural number. What are the possible cardinalities of point sets in a projective plane of order , which do not intersect any line at exactly points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this article, we show a series of results which point out the existence of all or almost all possible values for , provided that is not close to the extremal values 0 or . Moreover, using polynomial techniques we show that for every prescribed list of numbers , there exists a point set with the following property: holds for the th line , .

We study combinatorial parameters of a recently introduced bootstrap percolation problem in finite projective planes. We present sharp results on the size of the minimum percolating sets and the maximal nonpercolating sets. Additional results on the minimal and maximal percolation time as well as on the critical probability in the projective plane are also presented.

In the late 1950’s, B. Segre introduced the fundamental notion of arcs and complete arcs [48,49]. An arc in a finite projective plane is a set of points with no three on a line and it is complete if cannot be extended without violating this property. Given a projective plane $${\user1{P}}$$, determining $$n{\left( {\user1{P}} \right)}$$, the size of its smallest complete arc, has been a major open question in finite geometry for several decades. Assume that $${\user1{P}}$$ has order q, it was shown by Lunelli and Sce [41], more than 40 years ago, that $${\left( {\user1{P}} \right)} \geqslant {\sqrt {2q} }$$. Apart from this bound, practically nothing was known about $$n{\left( {\user1{P}} \right)}$$, except for the case $${\user1{P}}$$ is the Galois plane. For this case, the best upper bound, prior to this paper, was O(q3/4) obtained by Szőnyi using the properties of the Galois field GF(q).In this paper, we prove that $$n{\left( {\user1{P}} \right)} \leqslant {\sqrt q }{\kern 1pt} \log^{c} q$$ for any projective plane $${\user1{P}}$$ of order q, where c is a universal constant. Together with Lunelli-Sce’s lower bound, our result determines $$n{\left( {\user1{P}} \right)}$$ up to a polylogarithmic factor. Our proof uses a probabilistic method known as the dynamic random construction or Rodl’s nibble. The proof also gives a quick randomized algorithm which produces a small complete arc with high probability.The key ingredient of our proof is a new concentration result, which applies for non-Lipschitz functions and is of independent interest.

A problem originally suggested in the context of genetic coding leads naturally to the concept of {\em rook packing} and {\em error-distributing codes}. It is shown how various concepts in the theory of Latin squares, and also in coding theory, are best expressed in the form of questions about the placing of rooks on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> -dimensional hyperchessboards of side <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> . A new species of combinatorial design suggested by this is the concept of {\em optimal coloring}. It is shown that the optimal colorings in certain cases correspond to duals of desarguian projective planes. Light is thereby shed on the problems of the existence of both finite projective planes and close-packed single-error-correcting codes. In particular, the existence of a certain close-packed nonbinary single-error-correcting code, listed by Golay as the first unknown case, has been ruled out by a well-known result concerning Latin squares.

Exploring the classical Ceva configuration in a Desarguesian projective plane, we construct two families of minimal blocking sets as well as a new family of blocking semiovals in PG(2, 32h). Also, we show that these blocking sets of PG(2, q2), regarded as pointsets of the derived André plane , are still minimal blocking sets in . Furthermore, we prove that the new family of blocking semiovals in PG(2, 32h) gives rise to a family of blocking semiovals in the André plane as well.

We exhibit some techniques for constructing permutation arrays using projections in finite projective spaces and the geometry of arcs in the finite projective plane. We say a permutation array PA(n, d) has length n and minimum distance d when it consists of a collection of permutations on n symbols that pairwise agree in at most n − d coordinate positions. Such arrays can also be viewed as non-linear codes and are used in powerline communication. While our techniques likely do not produce optimal arrays, we are able to construct examples of codes for certain parameter sets for which no constructions were previously known.

Transitivity on Triangles in Finite Projective Planes

Simplicial complexes from finite projective planes and colored configurations