Finite Incidence Structures Research Articles

Light dual multinets of order six in the projective plane

The aim of this paper is twofold: First we classify all abstract light dual multinets of order $6$ which have a unique line of length at least two. Then we classify the weak projective embeddings of these objects in projective planes over fields of characteristic zero. For the latter we present a computational algebraic method for the study of weak projective embeddings of finite point-line incidence structures.

BRUCK NETS AND PARTIAL SHERK PLANES

In Bachmann [Aufbau der Geometrie aus dem Spiegelungsbegriff, Die Grundlehren der mathematischen Wissenschaften, Bd. XCVI (Springer, Berlin–Göttingen–Heidelberg, 1959)], it was shown that a finite metric plane is a Desarguesian affine plane of odd order equipped with a perpendicularity relation on lines and that the converse is also true. Sherk [‘Finite incidence structures with orthogonality’, Canad. J. Math.19 (1967), 1078–1083] generalised this result to characterise the finite affine planes of odd order by removing the ‘three reflections axioms’ from a metric plane. We show that one can obtain a larger class of natural finite geometries, the so-called Bruck nets of even degree, by weakening Sherk’s axioms to allow noncollinear points.

Constructions for key distribution patterns

Key distribution patterns (KDPs) are finite incidence structures satisfying a certain property which makes them widely used in minimizing the key storage and ensuring the security of communication between users in a large network. We construct a new KDP using t-design and combine two ω-KDPs to give new (ω-1)-KDPs, which provide secure communication in a large network and minimize the amount of key storage.

Linear closures of finite geometries

The interrelations between finite geometries (finite incidence structures) and linear codes over finite fields are discussed under some special fundamental aspects. For any incidence structure \({\mathcal{I}}\) block codes, block-difference codes and co-block codes over finite fields of characteristic p are discussed resp. introduced; correspondingly p-modular co-blocks are defined for \({\mathcal{I}}\). Orthogonality modulo p is introduced as a concept relating different geometries having the same point set. Conversely three types of block-tactical geometries may be derived from vector classes of fixed Hamming weight in a given linear code. These geometries are tactical configurations if the given code admits a transitive permutation group. A combination of both approaches leads to the concept of p-closure of a finite geometry and to the notions of p-closed, weakly p-closed and p-dense incidence structures. These geometric concepts are applied to simple or directed graphs via their natural “adjacency geometry”. Here the above mentioned code theoretic treatment leads to the concept of p-modular co-adjacent vertex sets. As instructive examples the Petersen graph, its complemetary graph and the Higman-Sims graph are considered.

Some Families of Directed Strongly Regular Graphs Obtained from Certain Finite Incidence Structures

We present many new directed strongly regular graphs by direct construction. We construct these graphs on the collections of antiflags of certain finite incidence structures. In this way, we confirm the feasibility of infinitely many parameter sets that was previously undetermined. We describe some examples of graphs together with their isomorphism classes to demonstrate the fact that our construction method is capable of producing many graphs with same parameters.

New invariants for incidence structures

We exhibit a new, surprisingly tight, connection between incidence structures, linear codes, and Galois geometry. To this end, we introduce new invariants for finite simple incidence structures \({\mathcal{D}}\), which admit both an algebraic and a geometric description. More precisely, we will associate one invariant for the isomorphism class of \({\mathcal{D}}\) with each prime power q. On the one hand, we consider incidence matrices M with entries from GF(qt) for the complementary incidence structure \({\mathcal{D}^*}\), where t may be any positive integer; the associated codes C = C(M) spanned by M over GF(qt); and the corresponding trace codes Tr(C(M)) over GF(q). The new invariant, namely the q-dimension\({{\rm dim}_q(\mathcal{D}^*)}\) of \({\mathcal{D}^*}\), is defined to be the smallest dimension over all trace codes which may be obtained in this manner. This modifies and generalizes the q-dimension of a design as introduced in Tonchev (Des Codes Cryptogr 17:121–128, 1999). On the other hand, we consider embeddings of \({\mathcal{D}}\) into projective geometries \({\Pi = PG(n, q)}\), where an embedding means identifying the points of \({\mathcal{D}}\) with a point set V in \({\Pi}\) in such a way that every block of \({\mathcal{D}}\) is induced as the intersection of V with a suitable subspace of \({\Pi}\). Our main result shows that the q-dimension of \({\mathcal{D}^*}\) always coincides with the smallest value of N for which \({\mathcal{D}}\) may be embedded into the (N − 1)-dimensional projective geometry PG(N − 1, q). We also give a necessary and sufficient condition when actually an embedding into the affine geometry AG(N − 1, q) is possible. Several examples and applications will be discussed: designs with classical parameters, some Steiner designs, and some configurations.

Some constructions for key distribution patterns

Key distribution patterns (KDPs) are finite incidence structures satisfying a certain property which enables them to be applied to a problem in network key distribution. Very few useful examples of KDPs are known. We discuss some ideas which can lead to constructions for KDPs and apply these ideas to construct infinite families of KDPs from conics in finite projective and affine planes.

New bounds and constructions for authentication/secrecy codes with splitting

We investigate authentication codes with splitting, using the mathematical model introduced by Simmons. Besides an overview of the existing bounds, we obtain some new bounds for the probability of deception of the transmitter/ receiver in case of an impersonation or substitution game. We also prove some new bounds for a “spoofing attack of order L.” Further, we give several new constructions for authentication/secrecy codes with splitting, derived from finite incidence structures such as partial geometries and affine resolvable designs. In some of these codes the bounds are attained with equality.

Near-rings (MDS)- and Laguerre codes

(MDS)- and Laguerre codes are closely related to geometry and can be used in order to construct certain finite incidence structures. Here we present some structure theorems on near rings, introduce the notion of a “coding set” of a near ring, which enables us to construct (MDS)-codes, and discuss the same problem for Laguerre codes. To find non trivial “Laguerre sets” in a near ring is much more difficult.

Intertwining automorphisms in finite incidence structures

The automorphism group of a finite incidence structure acts as permutation groups on the points and on the blocks of the structure. We view these actions as linear representations and observe that they are intertwined by the incidence relation. Most commonly the intertwining is of maximal linear rank, so that the representation on points appears as a subrepresentation of the action of the blocks. The paper investigates various consequences of this fact.

Compact disconnected planes, inverse limits and homomorphisms

Every compact disconnected projective plane can be written as an inverse limit of finite discrete incidence structures. Every finite projective plane is a continuous epimorphic image of some compact disconnected projective plane. There exist compact disconnected projective planes of Lenz type V which do not admit any continuous epimorphism onto a finite projective plane.

Key storage in secure networks

In this paper the key storage problem associated with the provision of secure communications between every pair of users in a large network is described, and a possible method of alleviating the problem is discussed. This method, based on the use of finite incidence structures with special properties called key distribution patterns, is shown to generalize earlier work in the area. The more general formulation of the storage saving scheme contained here enables use to made of the extensive body of knowledge already existing on the theory of block designs. From this theory we are able to extract a number of new families of examples of potentially useful key distribution systems.

Colouring finite incidence structures

One of our results: Let ℙ denote a finite projective plane of ordern. Colour its vertices byχ colours. If no colour appears more than twice on any of the lines, thenχ ≥ n + 1. Explicit constructions show that this bound is sharp when ℙ is desarguesian.

Rank 3 incidence structures admitting dual-linear, linear diagram

The results, restricted to finite Buekenhout incidence structures, are the following. (1) Let Γ be a finite rank 3 Buekenhout incidence structure admitting diagram ▪ Then for some generalized projective geometry π and some integer i, Γ is isomorphic to the rank 3 Buekenhout incidence structure having all ( i − 2)-, ( i − 1)-, and i-dimensional subspaces of π as varieties, and comparability as the incidence relation. (2) Let Γ be a finite rank 3 Buekenhout incidence structure admitting diagram ▪ Then for some set X and integer i, Γ is isomorphic to the rank 3 Buekenhout incidence structure having all ( i − 1)-, i-, and ( i + 1)-subsets of X as varieties, and set inclusion as the incidence relation.

Orbits in finite incidence structures

Orbits in finite incidence structures

Dembowski Semi-inversive planes

A complete classification has been made of all finite incidence structures of points and circles such that the residual structure with respect to an arbitrary point is always some Dembowski semi-plane.

An interactive program for finite incidence structures

It has been shown (see [3]) that every finite incidence structure can be represented in a suitable way by a polynomial over a convenient finite field. In this paper we present a FORTRAN interactive program which examines the incidence structure associated to a given polynomial over GF(9). Further, a second program is exhibited, which determines a complete system of mutually orthogonal latin squares related to a polynomial associated to a projective plane of order nine.

Finite Incidence Structures with Orthogonality

An incidence structure consists of two sets of elements, called points and blocks, together with a relation, called incidence, between elements of the two sets. Well-known examples are inversive planes, in which the blocks are circles, and projective and affine planes, in which the blocks are lines. Thus in various examples of incidence structures, the blocks may have various interpretations. Very shortly, however, we shall impose a condition (Axiom A) which ensures that the blocks behave like lines. In anticipation of this, we shall refer to the set of blocks as the set of lines. Also, we shall employ the usual terminology of incidence, such as “lies on,” “passes through,” “meet,” “join.” etc.