Applications of finite geometries to designs and codes

This dissertation concerns the intersection of three areas of discrete mathematics: finite geometries, design theory, and coding theory. The central theme is the power of finite geometry designs, which are constructed from the points and t-dimensional subspaces of a projective or affine geometry. We use these designs to construct and analyze combinatorial objects which inherit their best properties from these geometric structures. A central question in the study of finite geometry designs is Hamada’s conjecture, which proposes that finite geometry designs are the unique designs with minimum p-rank among all designs with the same parameters. In this dissertation, we will examine several questions related to Hamada’s conjecture, including the existence of counterexamples. We will also study the applicability of certain decoding methods to known counterexamples. We begin by constructing an infinite family of counterexamples to Hamada’s conjecture. These designs are the first infinite class of counterexamples for the affine case of Hamada’s conjecture. We further demonstrate how these designs, along with the projective polarity designs of Jungnickel and Tonchev, admit majority-logic decoding schemes. The codes obtained from these polarity designs attain error-correcting performance which is, in certain cases, equal to that of the finite geometry designs from which they are derived. This further demonstrates the highly geometric structure maintained by these designs. Finite geometries also help us construct several types of quantum error-correcting codes. We use relatives of finite geometry designs to construct infinite families of q-ary quantum stabilizer codes. We also construct entanglement-assisted quantum error-correcting codes (EAQECCs) which admit a particularly efficient and effective error-correcting scheme, while also providing the first general method for constructing these quantum codes with known parameters and desirable properties. Finite geometry designs are used to give exceptional examples of these codes

Relations among partitions

Combinatorialists often consider a balanced incomplete-block design to consist of a set of points, a set of blocks, and an incidence relation between them which satisfies certain conditions. To a statistician, such a design is a set of experimental units with two partitions, one into blocks and the other into treatments: it is the relation between these two partitions which gives the design its properties. The most common binary relations between partitions that occur in statistics are refinement, orthogonality and balance. When there are more than two partitions, the binary relations may not suffice to give all the properties of the system. I shall survey work in this area, including designs such as double Youden rectangles.PostprintPeer reviewe

Nearly Balanced and Resolvable Block Designs

One of the fundamental principles of experimental design is the separation of heterogeneous experimental units into subsets of more homogeneous units or blocks in order to isolate identifiable, unwanted, but unavoidable, variation in measurements made from the units. Given v treatments to compare, and having available b blocks of k experimental units each, the thoughtful statistician asks, “What is the optimal allocation of the treatments to the units?” This is the basic block design problem. Let nij be the number of times treatment i is used in block j and let N be the v x b matrix N = (nij). There is now a considerable body of optimality theory for block design settings where binarity (all nij ∈ {0, 1}), and symmetry or near-symmetry of the concurrence matrix NNT, are simultaneously achievable. Typically the same classes of designs are found to be best using any of the standard optimality criteria. Among these are the balanced incomplete block designs (BIBDs), many species of two-class partially balanced incomplete block designs, and regular graph designs. However, there are triples (v, b, k) in which binarity precludes near-symmetry. For these combinatorially problematic settings, recent explorations have resulted in new optimality results and insight into the combinatorial issues involved. Of particular interest are the irregular BIRD settings, that is, triples (v, b, k) where the necessary conditions for a BIBD are fulfilled but no such design exists. A thorough study of the smallest such setting, (15,21,5), has produced some surprising optimal designs which will be presented in the first chapter of this document. An incomplete block design is said to be resolvable if the blocks can be partitioned into classes, or replicates such that each treatment appears in exactly one block of each replicate. Resolvable designs are indispensable in many industrial and agricultural experiments, especially when the entire experiment can not be completed at one time or when there is a risk that the experiment may be prematurely terminated. In chapters two and three we will investigate the classes of resolvable designs having five or fewer replications and two blocks of possibly unequal size per replicate. Theory for identifying the best designs with respect to important optimality criteria will be developed, and with the optimality theory in hand, optimal designs will be identified and constructions provided. We will conclude with a comment on the robustness of resolvable designs to the loss of a replicate

Secure Protocols for Key Pre-distribution, Network Discovery, and Aggregation in Wireless Sensor Networks

The term sensor network is used to refer to a broad class of networks where several small devices, called sensors, are deployed in order to gather data and report back to one or more base stations. Traditionally, sensors are assumed to be small, low-cost, battery-powered, wireless, computationally constrained, and memory constrained devices equipped with some sort of specialized sensing equipment. In many settings, these sensors must be resilient to individual node failure and malicious attacks by an adversary, despite their constrained nature. This thesis is concerned with security during all phases of a sensor network's lifetime: pre-deployment, deployment, operation, and maintenance. This is accomplished by pre-loading nodes with symmetric keys according to a new family of combinatorial key pre-distribution schemes to facilitate secure communication between nodes using minimal storage overhead, and without requiring expensive public-key operations. This key pre-distribution technique is then utilized to construct a secure network discovery protocol, which allows a node to correctly learn the local network topology, even in the presence of active malicious nodes. Finally, a family of secure aggregation protocols are presented that allow for data to be efficiently collected from the entire network at a much lower cost than collecting readings individually, even if an active adversary is present. The key pre-distribution schemes are built from a family of combinatorial designs that allow for a concise mathematical analysis of their performance, but unlike previous approaches, do not suffer from strict constraints on the network size or number of keys per node. The network discovery protocol is focused on providing nodes with an accurate view of the complete topology so that multiple node-disjoint paths can be established to a destination, even if an adversary is present at the time of deployment. This property allows for the use of many existing multi-path protocols that rely on the existence of such node-disjoint paths. The aggregation protocols are the first designed for simple linear networks, but generalize naturally to other classes of networks. Proofs of security are provided for all protocols

Computational Statistics and Applications

Nature evolves mainly in a statistical way. Different strategies, formulas, and conformations are continuously confronted in the natural processes. Some of them are selected and then the evolution continues with a new loop of confrontation for the next generation of phenomena and living beings. Failings are corrected without a previous program or design. The new options generated by different statistical and random scenarios lead to solutions for surviving the present conditions. This is the general panorama for all scrutiny levels of the life cycles. Over three sections, this book examines different statistical questions and techniques in the context of machine learning and clustering methods, the frailty models used in survival analysis, and other studies of statistics applied to diverse problems

Geometry of symmetry

p. 325-412, [2] p. of plates : ill. ; 27 cm.Includes bibliographical references (p. 408-412)