Finite Desarguesian Planes Research Articles

Bounds for Arcs of Arbitrary Degree in Finite Desarguesian Planes

This paper examines subsets with at most n points on a line in the projective plane . A lower bound for the size of complete ‐arcs is established and shown to be a generalisation of a classical result by Barlotti. A sufficient condition ensuring that the trisecants to a complete (k, 3)‐arc form a blocking set in the dual plane is provided. Finally, combinatorial arguments are used to show that, for , plane (k, 3)‐arcs satisfying a prescribed incidence condition do not attain the best known upper bound.

Cubic Curves, Finite Geometry and Cryptography

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a ‘shared secret’ related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.

The transitive and co-transitive blocking sets in P^2(F_q)

We classify the transitive and co-transitive minimal blocking sets in a finite Desarguesian plane.

The correlations of finite Desarguesian planes, Part IV: The classification (III)

This paper is the last in a series of four articles devoted to the classification of correlations of finite Desarguesian planes of odd nonsquare order.

The correlations of finite Desarguesian planes of square order defined by diagonal matrices

The present article represents the next step in our ongoing program of classifying the correlations of finite Desarguesian planes. We show that in PG ( 2 , q 2 n ) , the correlations defined by diagonal matrices, with companion automorphism ( q m ), where ( m , 2 n ) = 1 , have the following numbers of absolute points: q 2 n + q n + 2 - q n + 1 + 1 or q 2 n - q n + 1 + q n + 1 or ( q n + 1 ) 2 for n odd; q 2 n - q n + 2 + q n + 1 + 1 or q 2 n + q n + 1 - q n + 1 or ( q n - 1 ) 2 for n even. We also discuss the equivalence classes into which these correlations fall, as well as the configurations of their sets of absolute points.

The correlations of finite Desarguesian planes, Part III: The classification (II)

The correlations of finite Desarguesian planes, Part III: The classification (II)

The correlations of finite Desarguesian planes, Part II: The classification (I)

The correlations of finite Desarguesian planes, Part II: The classification (I)

The correlations of finite Desarguesian planes, Part I: Generalities

The polarities of Desarguesian planes have long been known. This author has undertaken to classify the correlations of finite Desarguesian planes in general. In [6] we have presented all the correlations with identity companion automorphism which are not polarities, of these planes. In this sequence of papers, we classify the correlations of planes of order $ p^{2^{i}(2n+1)}, n \neq 0 $, with companion automorphism ( $p^{2^{i}t}$ ), p an odd prime, $ t \neq 0 $. This represents a complete classification of the correlations of planes of odd nonsquare order (i = 0). Some of the correlations of planes of odd square order ($ t \neq 0 $ ) are also covered by the present analysis.

The correlations with identity companion automorphism, of finite Desarguesian planes

As a first step towards the general classification of correlations of finite Desarguesian planes, we present, up to isomorphism, all the correlations with identity companion automorphism which are not polarities, of such planes.

Linear Perfect Codes and a Characterization of the ClassicalDesigns

A new definition for the dimension of a combinatorial t- (v,k,\lambda) design over a finite field is proposed. The complementary designs of the hyperplanes in a finite projective or affine geometry, and the finite Desarguesian planes in particular, are characterized as the unique (up to isomorphism) designs with the given parameters and minimum dimension. This generalizes a well-known characterization of the binary hyperplane designs in terms of their minimum 2-rank. The proof utilizes the q-ary analogue of the Hamming code, and a group-theoretic characterization of the classical designs.

Characterizations of Classical Algebraic Varieties in Some Papers of Giuseppe Tallini and Some Personal Memories of a Friend

One of the fundamental results of this century on finite geometries is undoubtedly the theorem by B.Segre characterizing rational points on a conic in finite Desarguesian planes of odd order q just as a set of q + 1 and three by three noncollinear points.Apart from its intrinsic scientific value, this celebrated and surprising result played a central role in the development of the geometry over Galois fields since it was the starting point of a new, wide and still active research field: to characterize rational points of classical algebraic varieties over finite fields in terms of their combinatorial and incidence properties. Giuseppe Tallini, who was among the favourite students of B.Segre and succeeded him as full professor at the University of Rome La Sapienza, was one of the leading geometers who gave great contributions to the above mentioned research area. I will now discuss the following Tallini's papers in this context.

On Conics that are Ovals in a Hall Plane

The conics of a finite Desarguesian plane of square even order satisfying the following properties are classified. (1) Their two infinite points (one being the nucleus) do not intersect a certain derivation set. (2) They are also ovals in the Hall plane constructed from the derivation set. This leads to a construction of ovals from conics in certain subregular planes of even order (which are translation planes of dimension 2 over their kernel).

The characterization of elliptic curves over finite fields

AbstractIn a finite Desarguesian plane of odd order, it was shown by Segre thirty years ago that a set of maximum size with at most two points on a line is a conic. Here, in a plane of odd or even order, sufficient conditions are given for a set with at most three points on a line to be a cubic curve. The case of an elliptic curve is of particular interest.

A proof of the theorem of Pappus in finite Desarguesian affine planes

Without using the representation theorem and a theorem of J.H.M. WEDDERBURN we show that every finite Desarguesian affine plane is Pappian.

A Beckman-Quarles type theorem for finite desarguesian planes

The characterization theorem given in [2] for Lorentz transformations of the ℝ2 is carried over to the case of finite planes.

On the algebras of symmetries (groups of collineations) of designs from finite Desarguesian planes with applications in statistics

Decomposition into a direct sum of irreducible representations of the representation of the full collineation group of a finite Desarguesian plane, as a group of matrices permuting the flags of the plane and the simple components of the corresponding commutant algebra, have been worked out here for the projective plane PG(2, 2) and the affine plane EG(2, 3). The dimension and the components of the covariance matrix of the observations from a design derived from such a plane, which commutes with such a permutation representation of the full collineation group of the plane, are thus determined. This paper is in the spirit of earlier works by, James (1957), Mann (1960), Hannan (1961, 1965), McLaren (1963), and Sysoev and Shaikan (1976). A. T. James, Ann. Math. Statist. 28 (1957), 993–1002, H. B. Mann, Ann. Math. Statist. 31 (1960), 1–15, E. J. Hannan, Research Report (Part. (I)), Summer Research Institute, Australian Math. Soc. and Methuen's Monographs on Applied Probability and Statistics, Supplementary Review Series in Applied Probability, Vol. 3, A. D. McLaren, Proc. Cambridge Philos. Soc. 59 (1963), 431–450, and L. P. Sysoev and M. E. Shaikin, Avtomat. i Telemekh. 5 (1976), 64–73.

On the algebras of symmetries (groups of collineations) of designs from finite Desarguesian planes with applications in statistics

On the algebras of symmetries (groups of collineations) of designs from finite Desarguesian planes with applications in statistics

On subplane partitions of a finite projective plane

Bruck (1960) notes that every finite Desarguesian plane of order m 2 can be partitioned into disjoint subplanes of order m. His method follows directly from the discovery by Singer (1938) that every such plane is cyclic and may be derived from a difference set. The present paper begins investigating the existence of other partitions which do not follow naturally from the difference set representation of the plane. The procedure is based on the idea, defined herein, of conjugacy with respect to a given subplane. When m = 2, only Bruck's partition is possible, but new results have been obtained in the cases m = 3, 4, 5, 7. This paper treats the case m = 3 in some detail.

Cyclic Affine Planes

Let Π be an affine plane which admits a collineation τ such that the cyclic group generated by τ leaves one point (say X) fixed, and is transitive on the set of all other points of Π. Such “cyclic affine planes” have been previously studied, especially in India, and the principal result relevant to the present discussion is the following theorem of Bose [2]: every finite Desarguesian affine plane is cyclic. The converse seems quite likely true, but no proof exists. In what follows, we shall prove several properties of cyclic affine planes which will imply that for an infinite number of values of n there is no such plane with n points on a line.