Group Embedding of the Projective Plane PG(2, 3)

The functions a(n) and p(n) are defined to be the smallest integer λ for which λ‐fold quasimultiples affine and projective planes of order n exist. It was shown by Jungnickel [J. Combin. Designs 3 (1995), 427–432] that a(n),p(n) < n10 for sufficiently large n. In the present paper, we prove that a(n),p(n) < n3. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 182–186, 2001

Let Π be a projective plane of order n admitting a collineation group G≅PSL(2, q) for some prime power q. It is well known for n=q that Π must be Desarguesian. We show that if n<q then only finitely many cases may occur for П, all of which are Desarguesian. We obtain some information in case n=q2 with q odd, notably that G acts irreducibly on П for q≠3, 5, 9.

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.

On defining sets for projective planes

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 , .

A Combinatorial Distinction Between the Euclidean and Projective Planes

An optimal bound for embedding linear spaces into projective planes

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].

Reduction of symmetric configurations n3

The structure of a collineation group G preserving a unital in a projective plane of order m2 with m ≡ 1(4) is investigated and several results are obtained about G, when G is a 2-group. These results are then used to investigate the structure of G/O(G) in the general case and, in particular, that of F*(G/O(G)).

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 \(\mathrm{PG}(2, q)\).

We give a short proof for the existence of the small Witt design which is\nbased on the projective plane of order three with one point deleted.\n

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 \} $.

This thesis consists of two independent chapters. The first chapter concerns the Smith Normal Form (SNF) over the integers ℤ of integral matrices. We consider the SNF of a matrix A to be the ratio of two ℤ-modules -- a finitely generated abelian group; this is called the Smith group of A. The Smith group provides a unified setting to present both new and old results. The new results concern the relationship between the eigenvalues of an integral matrix and its SNF. In particular, the multiplicities of integer eigenvalues are shown to relate to the multiplicities in the type of the Smith group. Bounds are also given for the exponent of the Smith group. In some cases, these are best possible. The old results discussed are the interlacing of the SNF in the case of augmented matrices and the symmetries of the SNF for certain combinatorial matrices. The latter results are extended to rectangular matrices. Numerous examples are given throughout, along with many conjectures based on computation. The second chapter generalizes the work of Pless, et al. on duadic codes and Q-codes. We take abelian group codes to be ideals in the group ring F[G], where G is a finite abelian group of odd order n and F is a finite field with characteristic relatively prime to n. We define generalized Q-codes from a pair of idempotents of F[G] and an automorphism of G which together obey two simple equations. These codes are (n, (n+1)/2) and (n, (n-1)/2) linear codes. We show that all of the properties of duadic and Q-codes generalize. In particular, we extend the results on the relationship of these codes to projective planes with regular automorphism group G. When F has characteristic 2, we give simple numerical conditions on G and F which determine when generalized Q-codes exist. We also give some techniques for constructing these codes.

We consider a class of symmetric divisible designs D which are almost projective planes in the following sense: Given any point p, there is a unique point p’ such that p and p’ are on two lines, whereas any other point is joined to p by exactly one line; and dually. We note that either the block size k or k — 2 is a perfect square, and exhibit examples for k =3 and k = 4. Then we add the condition that D should admit an abelian Singer group, so that we may study the associated divisible difference sets. Under this additional assumption, we show that k is a square (unless k = 3) and that the only possible prime divisors of k are 2 and 3.