On finite moore geometries

Conditions are given which must be satisfied by the parameters of a Moore geometry. It is shown that there are no nontrivial Moore geometries of diameter 3.     

Semigroup closures of finite rank symmetric inverse semigroups

We introduce the notion of semigroup with a tight ideal series and investigate their closures in semitopological semigroups, particularly inverse semigroups with continuous inversion. As a corollary we show that the symmetric inverse semigroup of finite transformations I _λ ⁿ of the rank ≤ n is algebraically closed in the class of (semi)topological inverse semigroups with continuous inversion. We also derive related results about the nonexistence of (partial) compactifications of classes of semigroups that we consider.     

Minimal balanced sets and finite geometries

A balanced set is a collection of subsets of a finite set that can be weighted so as to cover the whole set uniformly. Minimal balanced sets are of interest in the theory of n-person games, in particular for the existence of outcomes that cannot be improved upon by any coalition (core of the game).The object of this paper is to determine the finite geometries which are minimal balanced sets. We prove that the dual of any t-design with t ? 2 is a minimal balanced set. In particular symmetrical 2-designs (as projective spaces, biplanes, etc.) are always minimal balanced sets. For 1-designs the problem becomes much more difficult, but it is for instance easy to prove that any partial geometry which is not the dual of a 2-Steiner system is never a minimal balanced set; in particular generalized quadrangles are never minimal balanced sets. For linear graphs the problem is completely solved: the dual of a connected linear graph is a minimal balanced set if and only if this linear graph is not bichromatic.     

Partitions of finite affine geometries into caps

In a finite geometry of order q² we define a (q^mq^r)-affine cap to be a set of cardinality q^m which is a disjoint union ot q^m affine subgeometrics AG(r,q). such that no three points are coliinear unless contained in the same AG(r,q).

Given a PG(n,q²), where n = 2t + 1 or 2t + 2, and an n + 1 by n + 1 Hermitian matrix H over Gh(q²) with minimal polynomial (x - λ)^{n + 1}. we show that H induces a partition of the AG(n, q²) obtained by deleting a distinguished hyperplane from the PG, into (qⁿ,q^{l + 1})-affine caps; these caps can be viewed as the "large points" of an AG (n,q) with a natural incidence relation. It is also shown that H induces another partition of AG(n,q²), into q^{n - l 1}-caps, constituting the "large points" of an affine geometry AG(n + t + 1,q).

Also, the collineation C of PG(n, q²) given by x^c = H^Tx induces collineations on the AG(n,q) and AG(n + t + 1,q).     

Integral closures and weight functions over finite fields

Curves and surfaces of type I are generalized to integral towers of rank r. Weight functions with values in N^r and the corresponding weighted total-degree monomial orderings lift naturally from one domain R_j−1 in the tower to the next, R_j, the integral closure of R_j−1[x_j]/φ(x_j). The qth power algorithm is reworked in this more general setting to produce this integral closure over finite fields, though the application is primarily that of calculating the normalizations of curves related to one-point AG codes arising from towers of function fields. Every attempt has been made to couch all the theory in terms of multivariate polynomial rings and ideals instead of the terminology from algebraic geometry or function field theory, and to avoid the use of any type of series expansion.     

Some classes of finite homomorphism-homogeneous point-line geometries

In 2006, P. J. Cameron and J. Ne?et?ril introduced the following variant of homogeneity: we say that a structure is homomorphism-homogeneous if every homomorphism between finite substructures of the structure extends to an endomorphism of the structure. In this paper we classify finite homomorphism-homogeneous point-line geometries up to a certain point. We classify all disconnected point-line geometries, and all connected point-line geometries that contain a pair of intersecting proper lines (we say that a line is proper if it contains at least three points). In a way, this is the best one can hope for, since a recent result by Rusinov and Schweitzer implies that there is no polynomially computable characterization of finite connected homomorphism-homogeneous point-line geometries that do not contain a pair of intersecting proper lines (unless P=coNP).     

Designs and partial geometries over finite fields

Let be a finite field, and let (, B) be a nontrivial 2-(n, k, 1)-design over . Then each point induces a (k–1)-spread S on /. (, B) is said to be a geometric design if S is a geometric spread on / for each . In this paper, we prove that there are no geometric designs over any finite field .Research partially supported by NSF grant DMS-8703229.     

Nonbinary quantum codes derived from finite geometries

The paper gives explicit parameters for several infinite families of q-ary quantum stabilizer codes. These codes are derived from combinatorial designs which arise from finite projective and affine geometries.     

A characterization of point-line geometries for finite buildings

We present an axiomsystem for almost (all) point-line geometries associated to finite buildings (where the j-varieties, for certain j, are called points) containing a polar subspace of rank at least three (see list in addition), and we develop methods of construction to attain that goal.     

Some low-density parity-check codes derived from finite geometries

We look at low-density parity-check codes over a finite field \mathbbK{\mathbb{K}} associated with finite geometries T₂^*(K){T_2^*(\mathcal{K})}, where K{\mathcal{K}} is a sufficiently large k-arc in PG(2, q), with q = p ^h . The code words of minimum weight are known. With exception of some choices of the characteristic of \mathbbK{\mathbb{K}} we compute the dimension of the code and show that the code is generated completely by its code words of minimum weight.     

Linear independence in finite spaces

The maximum number m ₂(n, q) of points in PG(n, q), n2, such that no three are collinear is known precisely for (n, q)=(n,2), (2,q), (3,q), (4, 3), (5,3). In this paper an improved upper bound of order q ^n–1 –1/2q ^n–2 is obtained for q even when n4 and q>2. A necessary preliminary is an improved upper bound for m₂(3, q), the maximum size of a k-cap not contained in an ovoid. It is shown that and that m₂(3, 4)=14.     

Characterization of chain geometries of finite dimension by their automorphism group

A large class of chain geometries of finite dimension is characterized as strong chain spaces possessing a distinguished group of automorphisms fixing two distant points.Herrn Professor Dr. H. Salzmann zum 60. Geburtstag