Extendable Steiner designs from finite geometries

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.

Open problems in finite projective spaces

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.

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.

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

Triangle presentations are combinatorial structures on finite projective geometries which characterize groups acting simply transitively on the vertices of a locally finite building of type \tilde{A}_{n-1},\ (n \ge 3) . From a type \tilde{A}_{n-1} triangle presentation on a geometry of order q , we construct a fiber functor on the diagrammatic monoidal category \mathrm{Web}(\mathrm{SL}^{-}_{n}) over any field \mathbb{k} with characteristic p\ge n − 1 such that q \equiv 1\ \mathrm{mod}\ p . When \mathbb{k} is algebraically closed and n odd, this gives new fiber functors on the category of tilting modules for \mathrm{SL}_{n} .

In this paper, we investigate the properties of the point-line incidence matrix of a finite affine plane and present a construction of quasicyclic LDPC codes based on submatrices of the incidence matrix. The codes from the point-line incidence matrices of the affine and projective geometries perform very well with iterative decoding algorithms for low-density parity-check (LDPC) codes.

Codes C 1 ,…,C M of length n over ? q and an M × N matrix A over ? q define a matrix-product code C = [C 1 …C M ] ·A consisting of all matrix products [c 1 … c M ] ·A. This generalizes the (u|u+v)-, (u+v+w|2u+v|u)-, (a+x|b+x|a+b+x)-, (u+v|u-v)- etc. constructions. We study matrix-product codes using Linear Algebra. This provides a basis for a unified analysis of |C|, d(C), the minimum Hamming distance of C, and C ⊥. It also reveals an interesting connection with MDS codes. We determine |C| when A is non-singular. To underbound d(C), we need A to be `non-singular by columns (NSC)'. We investigate NSC matrices. We show that Generalized Reed-Muller codes are iterative NSC matrix-product codes, generalizing the construction of Reed-Muller codes, as are the ternary `Main Sequence codes'. We obtain a simpler proof of the minimum Hamming distance of such families of codes. If A is square and NSC, C ⊥ can be described using C 1 ⊥, …,C M ⊥ and a transformation of A. This yields d(C ⊥). Finally we show that an NSC matrix-product code is a generalized concatenated code.

On monomial codes in modular group algebras

A dual-protection approach to enhance optical image encryption using QZ decomposition and generalized Reed–Muller codes

Several combinatorial properties of low-density parity-check (LDPC) codes, such as minimum distance, diameter, stopping number, girth and cycle-length distribution of the corresponding Tanner graph, are known to influence their performance under iterative decoding. Recently, a new class of combinatorial configurations, termed trapping sets, was shown to be of significant importance in determining the properties of LDPC codes in the error-floor region. Very little is known both about the existence/parameters of trapping sets in structured LDPC codes and about possible techniques for reducing their negative influence on the code's performance. In this paper, we address both these problems from an algorithmic and combinatorial perspective. We first provide a numerical study of the trapping phenomena for the Margulis code, which exhibits a fairly high error-floor. Based on this analysis, conducted for two different implementations of iterative belief propagation, we propose a novel decoding process, termed averaged decoding. Averaged decoding provides for a significant reduction in the number of incorrectly decoded frames in the error-floor region of the Margulis code. Furthermore, based on the results of the algorithmic approach, we suggest a novel combinatorial characterizations of trapping sets in the class of LDPC codes based on finite geometries. Projective geometry LDPC codes are suspected to have extremely low error-floors, which is a property that we may attribute to the non-existence of certain small trapping sets in the code graph.