In this dissertation we present new results regarding optimality of block designs with limited resources. The dissertation is organized as follows. The first chapter outlines the theory of optimal block design. The second chapter shows new work in optimal minimally connected block designs with spatial correlation structure. The third chapter details the discovery of the optimal incomplete designs with two blocks. The fourth chapter does the same for the optimal binary incomplete designs with three blocks. The fifth chapter summarizes the techniques used and new results found and lists possible future research topics. / Ph. D.

We consider a class of optimization problems in which the aim is to find some optimizing probability distributions. One particular example is optimal design. We first review the optimal design theory, and determine the optimality conditions using directional derivatives. We then construct optimal designs for various polynomial regression models by finding the analytic solutions and by using a class of algorithms. We consider a practical problem, namely a radiation-dosage example, and discuss important aspects of optimal design throughout this example. We also construct optimal designs for various polynomial regression models with more than one design variable. We consider another practical problem, namely a vocabulary-growth study. We then construct D-optimal and c-optimal designs for various models with and without the interaction term and the second order terms in design variables. We also develop strategies for constructing designs by using the properties of the directional derivatives.

We consider a class of optimization problems in which the aim is to find some optimizing probability distributions. One particular example is optimal design. We first review the optimal design theory, and determine the optimality conditions using directional derivatives. We then construct optimal designs for various polynomial regression models by finding the analytic solutions and by using a class of algorithms. We consider a practical problem, namely a radiation-dosage example, and discuss important aspects of optimal design throughout this example. We also construct optimal designs for various polynomial regression models with more than one design variable. We consider another practical problem, namely a vocabulary-growth study. We then construct D-optimal and c-optimal designs for various models with and without the interaction term and the second order terms in design variables. We also develop strategies for constructing designs by using the properties of the directional derivatives.

A double-change covering design (dccd) is an ordered set of blocks with block size k is an ordered collection of b blocks, B = {B1,B2, · · · ,Bb}, each an unordered subset of k distinct elements from [v] = {1, 2, · · · , v}, which obey: (1) each block differs from the previous block by two elements, and, (2) every unordered pair of [v] appears in at least one block. The object is to minimize b for a fixed v and k. Tight designs are those in which each pair is covered exactly once. We present constructions of tight dccd’s for arbitrary v when k = 2 and minimal constructions for v <= 20 when k = 4. A general, but not minimal, method is presented to construct circular dccd for arbitrary v when k = 4.

The growing complexity of System-on-Chip (SoC) designs motivates both academic and industrial researchers to find better solutions for the complexity of the chip-interconnect. For SoC designs that have hundreds of Processing Elements (PEs), a single shared bus can no longer be accepted as an efficient communication scheme. To address this problem, the Networks-on-Chip (NoC) concept is proposed as a new paradigm, which provides an integrated solution for achieving efficient interconnection scheme for complex SoC applications. NoC-based designs are composed of computational resources in the form of PE cores, and switching nodes (routers) that allow PEs to communicate with each other. For different applications, this research work: 1) proposes new analytical models for various NoC design parameters, 2) performs comparative analyses of the commonly used network architectures, and 3) presents novel methodologies for efficiently designing the NoC-topology. The proposed methodologies are developed to help NoC-designers better achieve minimum power consumption and delay, and maximum performability for their applications. Graph-theoretic concepts are adopted to study the topological architecture of NoCs and propose a new topology-based models for network power, performability, and delay. The proposed models take into consideration important design parameters, which significantly affect the power, performability, and delay of a NoC-based system; such as network topology architecture, traffic distribution, noise power, voltage swing, probability of edge failure, router design and number of ports, clock frequency, and target technology. In this dissertation, we show how the proposed models could be used to optimally design the network topology so that it achieves the target design requirement for a given application. After studying each design metric individually, a joint consideration of NoC power, performability, and delay is carried out simultaneously. We use Particle Swarm Optimization (PSO) to find the optimum network topology, that achieves minimum delay, maximum performability, and minimum power consumption, for a given NoC application. Real case studies are presented to validate the proposed theoretical concepts. This validation is carried out through experimental work, targeting various real NoC applications. Experimental results show that using the proposed design methodologies, designers can improve the overall system efficiency in terms of power, delay, and performability, by choosing the design parameters (i.e., network topology architecture, PEs’ mapping, etc.) efficiently at early design phases. This improvement is measured in some cases by an order of magnitude, compared to the worst case scenario of choosing wrong design parameters for the target application. / Graduate

A balanced tournament design of order <I>n</I>, BTD(<I>n</I>), defined on a 2<I>n</I>-set<I> V</i>, is an arrangement of the all of the (2<I>n</i>2) distinct unordered pairs of elements of <I>V</I> into an <I>n</I> X (2<I>n</i> - 1) array such that (1) every element of <I>V</i> occurs exactly once in each column and (2) every element of <I>V</I> occurs at most twice in each row. We will show that there exists a BTD(<i>n</i>) for <i>n</i> a positive integer, <i>n</i> not equal to 2. For <I>n</i> = 2, a BTD (<i>n</i>) does not exist. If the BTD(<i>n</i>) has the additional property that it is possible to permute the columns of the array such that for every row, all the elements of<I> V</I> appear exactly once in the first <i>n</i> pairs of that row and exactly once in the last <i>n</i> pairs of that row then we call the design a partitioned balanced tournament design, PBTD(<I>n</I>). We will show that there exists a PBTD (<I>n</I>) for <I>n</I> a positive integer, <I>n</I> is greater than and equal to 5, except possibly for <I>n</I> an element of the set {9,11,15}. For <I>n</I> less than and equal to 4 a PBTD(<I>n</I>) does not exist.

A balanced tournament design of order <I>n</I>, BTD(<I>n</I>), defined on a 2<I>n</I>-set<I> V</i>, is an arrangement of the all of the (2<I>n</i>2) distinct unordered pairs of elements of <I>V</I> into an <I>n</I> X (2<I>n</i> - 1) array such that (1) every element of <I>V</i> occurs exactly once in each column and (2) every element of <I>V</I> occurs at most twice in each row. We will show that there exists a BTD(<i>n</i>) for <i>n</i> a positive integer, <i>n</i> not equal to 2. For <I>n</i> = 2, a BTD (<i>n</i>) does not exist. If the BTD(<i>n</i>) has the additional property that it is possible to permute the columns of the array such that for every row, all the elements of<I> V</I> appear exactly once in the first <i>n</i> pairs of that row and exactly once in the last <i>n</i> pairs of that row then we call the design a partitioned balanced tournament design, PBTD(<I>n</I>). We will show that there exists a PBTD (<I>n</I>) for <I>n</I> a positive integer, <I>n</I> is greater than and equal to 5, except possibly for <I>n</I> an element of the set {9,11,15}. For <I>n</I> less than and equal to 4 a PBTD(<I>n</I>) does not exist.

When defining a structure to fulfill a set of axioms that are similar to those prescribed by Euclid, one must select a set of points and then define what is meant by a line and what is meant by a circle. When properly defined these labels will have properties which are similar to their counterparts in the (complex) plane, the lines and circles which Euclid undoubtedly had in mind. In this manner, the geometer may employ his intuition from the complex plane to prove theorems about other systems. Most "finite geometries" have clearly defined notions of points and lines but fail to define circles. The two notable exceptions are the circles in a finite affine plane and the circles in a Mobius plane. Using the geometry of Euclid as motivation, we strive to develop structures with both lines and circles. The only successful example other than the complex plane is the affine plane over a finite field, where all of Euclid's geometry holds except for any assertions involving order or continuity. To complement the prolific work concerning finite geometries and their lines, we provide a general definition of a circle, or more correctly, of a collection of circles and present some preliminary results concerning the construction of such structures. Our definition includes the circles of an affine plane over a finite field and the circles in a Mobius plane as special cases. We develop a necessary and sufficient condition for circularity, present computational techniques for determining circularity and give varying constructions. We devote a chapter to the use of circular designs in coding theory. It is proven that these structures are not useful in the theory of error-correcting codes, since more efficient codes are known, for example the Reed-Muller codes. However, the theory developed in the earlier chapters does have applications to Cryptology. We present five encryption methods utilizing circular structures.

It was not until the 20th century that combinatorial design theory was studied as a formal subject. This field has many applications, for example in statistical experimental design, coding theory, authentication codes, and cryptography. Major approaches to the problem of discovering new t-designs rely on (i) the construction of large sets of t designs, (ii) using prescribed automorphism groups, (iii) recursive construction methods. In 2017 and 2018, Tran Van Trung introduced new recursive techniques to construct t – (v, k, λ) designs. These methods are of purely combinatorial nature and require using "ingredient" t-designs or resolutions whose parameters satisfy a system of non-linear equations. Even after restricting the range of parameters in this new method, the task is computationally intractable. In this work, we enhance Tran Van Trung's "Basic Construction" by a robust and efficient hybrid computational apparatus which enables us to construct hundreds of thousands of new t – (v, k, Λ) designs from previously known ingredient designs. Towards the end of the dissertation we also create a new family of 2-resolutions, which will be infinite if there are infinitely many Sophie Germain primes. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

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