Triangle presentations and tilting modules for SL$_{2k+1}$

In recent years there has been an increasing interest in finite projective spaces, and important applications to practical topics such as coding theory and design of experiments have made the field even more attractive.It is my intention to mention some important and elegant theorems, to say something about the used techniques and the relation with other fields, and to mention some open problems.First some characterizations of particular pointsets in the projective space PG(n, q), n 2, over GF(q) will be given, where, from the beginning, it is assumed that the pointset is contained in PG(n, q).A second approach is that where the object is described as an incidence structure satisfying certain properties; here the geometry is not a priori embedded in a projective space.This approach will be illustrated with some theorems on inversive planes, polar spaces and Shult spaces.Finally, there is a section on k-arcs in PG(n, q) and on linear Maximum Distance Separable codes, where the interplay between finite projective geometry, coding theory and algebraic geometry is particularly present.In an appendix an example of brand new research in the field is given.

Design of a balanced multiple-valued filing scheme of order two based on cyclically generated spread in finite projective geometry

In the first part of this paper there are found the numbers of points, lines, etc., in a finite projective geometry of n dimensions. The substance of this has already been worked out by O. Veblen and W. H. Bussey. The second part is concerned with the arrangements of the numbers representing the points in a finite projective plane desarguesian geometry.

Extendable Steiner designs from finite geometries

Open problems in finite projective spaces

For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding. However, this notion requires additional structure to be present, namely a fiber functor. For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties—expressible entirely within the language of tensor categories—which are necessary and sufficient for the existence of a fiber functor. In this paper we generalize Deligne’s result to weakly Tannakian categories in characteristic zero. The class of geometric objects whose tensor categories of quasi-coherent sheaves can be recognized in this manner includes both the gerbes arising in classical Tannaka duality and more classical geometric objects such as projective varieties over a field of characteristic zero. Our proof uses a different perspective on fiber functors, which we formalize through the notion of geometric tensor categories. A second application of this perspective allows us to describe categories of quasi-coherent sheaves on fiber products.

Dey and Suen (2002) obtained optimal fractional factorial plans for estimating mean, all main effects and specified two-factor interactions using finite projective geometry. In this article we obtain two new optimal fractional factorial plans for estimation of the mean, all main effects, specified two-factor interactions,… and specified (r − 1) factor interactions using PG(r − 1, m). We also provide tables of optimal fractional factorial plans.

Decomposable symmetric designs

A line in $({\Bbb Z}/n{\Bbb Z})^2$ is any translate of a cyclic subgroup of order $n$. A subset $X\subset ({\Bbb Z}/n{\Bbb Z})^2$ is a cap if no three of its points are collinear, and $X$ is complete if it is not properly contained in another cap. We determine bounds on $\Phi(n)$, the minimum size of a complete cap in $({\Bbb Z}/n{\Bbb Z})^2$. The other natural extremal question of determining the maximum size of a cap in $({\Bbb Z}/n{\Bbb Z})^2$ is considered in a separate preprint by the present author. These questions are closely related to well-studied questions in finite affine and projective geometry. If $p$ is the smallest prime divisor of $n$, we prove that $$\max\{4,\sqrt{2p}+{1\over2}\}\leq \Phi(n)\leq \max\{4,p+1\}.$$ We conclude the paper with a large number of open problems in this area.

A well-known conjecture of András Gyárfás and David Sumner states that for every positive integer m and every finite tree T there exists k such that all graphs that do not contain the clique Km or an induced copy of T have chromatic number at most k. The conjecture has been proved in many special cases, but the general case has been open for several decades. The main purpose of this paper is to consider a natural analogue of the conjecture for matroids, where it turns out, interestingly, to be false. Matroids are structures that result from abstracting the notion of independent sets in vector spaces: that is, a matroid is a set M together with a nonempty hereditary collection I of subsets deemed to be independent where all maximal independent subsets of every set are equicardinal. They can also be regarded as generalizations of graphs, since if G is any graph and I is the collection of all acyclic subsets of E(G), then the pair (E(G),I) is a matroid. In fact, it is a binary matroid, which means that it can be represented as a subset of a vector space over F2. To do this, we take the space of all formal sums of vertices and represent the edge vw by the sum v+w. A set of edges is easily seen to be acyclic if and only if the corresponding set of sums is linearly independent. There is a natural analogue of an induced subgraph for matroids: an induced restriction of a matroid M is a subset M′ of M with the property that adding any element of M−M′ to M′ produces a matroid with a larger independent set than M′. The natural analogue of a tree with m edges is the matroid Im, where one takes a set of size m and takes all its subsets to be independent. (Note, however, that unlike with graph-theoretic trees there is just one such matroid up to isomorphism for each m.) Every graph can be obtained by deleting edges from a complete graph. Analogously, every binary matroid can be obtained by deleting elements from a finite binary projective geometry, that is, the set of all one-dimensional subspaces in a finite-dimensional vector space over F2. Finally, the analogue of the chromatic number for binary matroids is a quantity known as the critical number introduced by Crapo and Rota, which in the case of a graph G turns out to be ⌈log2(χ(G))⌉ -- that is, roughly the logarithm of its chromatic number. One of the results of the paper is that a binary matroid can fail to contain I3 or the Fano plane F7 (which is the simplest projective geometry) as an induced restriction, but also have arbitrarily large critical number. By contrast, the critical number is at most two if one also excludes the matroid associated with K5 as an induced restriction. The main result of the paper is a structural description of all simple binary matroids that have neither I3 nor F7 as an induced restriction.

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.

ABSTRACT: In this paper, coexistence of and relations between association schemes, orthogonal arrays and certain families of projective codes have been examined. The projective codes considered are linear spans of a nice projective set P in a hyperplane H = PG ( N -1, s)-such as a quadric or a quadric with its nucleus of polarity or a Hermitian variety. There are two ways to construct association schemes from a projective code. One due to Delsarte (1973) considers the restriction of the Hamming scheme to the code with m weights and if it satisfies Delsarte's condition, an m-class association scheme is obtained by defining two codewords to be i-th associates if the Hamming distance between them is i, i = 0, 1, …, m. The alternative approach, first used by Ray-Chaudburi (1959) and later generalized by Mesner (1967) is to classify points (according to some geometrical criterion) in H = PG(N-1, s) with reference to P. into m types (say). Then, two points of the affine space EG( N, s) (for which H is the hyperplane at infinity) are defined to be i-th associates if the line joining the two points meet H at a point of type i, i = 1, …, m. In many cases, the two association schemes defined with respect to the same projective set have the same parameters. But Examples are given where they do not coincide and, in fact, there are cases where one scheme exists but the other does not.

Finite convex geometries are combinatorial structures. It follows from a recent result of M. Richter and L.G. Rogers that there is an infinite set $T_{rr}$ of planar convex polygons such that $T_{rr}$ with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of $T_{rr}$ to a finite subset in a natural way. An \emph{almost-circle of accuracy} $1-\epsilon$ is a differentiable convex simple closed curve $S$ in the plane having an inscribed circle of radius $r_1>0$ and a circumscribed circle of radius $r_2$ such that the ratio $r_1/r_2$ is at least $1-\epsilon$. % Motivated by Richter and Rogers' result, we construct a set $T_{new}$ such that (1) $T_{new}$ contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves; (2) $T_{new}$ with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to $T_{rr}$, $T_{new}$ is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive $\epsilon\in\real $ and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets $E$ of $T_{new}$ such that each $E$ consists of almost-circles of accuracy $1-\epsilon$ and the convex geometry in question is represented by restricting the convex hull operator to $E$. The affine-disjointness of $E_1$ and $E_2$ means that, in addition to $E_1\cap E_2=\emptyset$, even $\psi(E_1)$ is disjoint from $E_2$ for every non-degenerate affine transformation $\psi$.

The weight hierarchy of a linear [n; k; q] code C over GF(q) is the sequence (d1, d2, ···, dk) where dr is the smallest support of any r-dimensional subcode of C. “Determining all possible weight hierarchies of general linear codes” is a basic theoretical issue and has important scientific significance in communication system. However, it is impossible for q-ary linear codes of dimension k when q and k are slightly larger, then a reasonable formulation of the problem is modified as: “Determine almost all weight hierarchies of general q-ary linear codes of dimension k”. In this paper, based on the finite projective geometry method, the authors study q-ary linear codes of dimension 5 in class IV, and find new necessary conditions of their weight hierarchies, and classify their weight hierarchies into 6 subclasses. The authors also develop and improve the method of the subspace set, thus determine almost all weight hierarchies of 5-dimensional linear codes in class IV. It opens the way to determine the weight hierarchies of the rest two of 5-dimensional codes (classes III and VI), and break through the difficulties. Furthermore, the new necessary conditions show that original necessary conditions of the weight hierarchies of k-dimensional codes were not enough (not most tight nor best), so, it is important to excogitate further new necessary conditions for attacking and solving the k-dimensional problem.

This paper proposes a recursive method for constructing intra-resolvable balanced incomplete block designs (BIBDs). The approach exploits the algebraic and geometric structure of finite projective geometries over Galois fields to generate resolvable designs with improved efficiency in terms of the number of blocks and treatment replications. The recursive procedure produces symmetric and uniform designs that are particularly suitable for high-dimensional settings. By systematically nesting resolvable blocks, we derive a new class of balanced n-ary designs that are both economical and scalable. These designs hold significant value for the statistical community, offering broad applicability in resource-constrained experimental environments such as precision agriculture, high-throughput drug screening, and computer-based simulation studies. We provide theoretical foundations through explicit constructions and comparative evaluations, demonstrating the advantages of our method over classical approaches.