Dissertations / Theses on the topic 'Balanced Incomplete Block Design'

Nonparametric tests are used to test hypotheses when the data at hand violate one or more of the assumptions for parametric tests procedures. The test is an ordered alternative (nondecreasing) when there is prior information about the data. It assumes that the underlying distributions are of the same type and therefore differ in location. For example, in dose-response studies, animals are assigned to k groups corresponding to k doses of an experimental drug. The effect of the drug on the animals is likely to increase or decrease with increasing doses. In this case, the ordered alternative is appropriate for the study. In this paper, we propose eight new nonparametric tests useful for testing against nondecreasing order alternatives for a mixed design involving randomized complete block and balanced incomplete block design. These tests involve various modifications of the Jonckheere-Terpstra test (Jonckheere(1952), Terpstra(1954)) and Alvo and Cabilio’s test (1995). Three, four and five treatments were considered with different location parameters under different scenarios. For three and four treatments, 6,12, and 18 blocks were used for the simulation, while 10, 20, and 30 blocks were used for five treatments. Different tests performed best under different block combinations, but overall the standardized last for Alvo outperformed the other test when the number of treatments and number of missing observations per block increases. A simulation study was conducted comparing the powers of the various modification of Jonckheere-Terpstra (Jonckheere(1952), Terpstra(1954)) and Alvo and Cabilio’s (1995) tests under different scenarios. Recommendations are made.

The ever-increasing demand and requirements of communication systems necessitates research into advanced techniques to ensure reliable and optimum use of communication resources such as bandwidth. Channel characteristics such as noise, erasures and fading have a destructive effect on the transmission requiring novel approaches to obtain a reliable reconstruction at the receiver. In this thesis, we investigate error protection using linear block codes for Gaussian noise, erasure and fading channels. In the design of an error protection system, we focus on a class of near Shannon-limit approaching codes called Low-Density Parity-Check (LDPC) codes. The work presents and investigates a method of constructing LDPC codes using combinatorial mathematics known as Balanced Incomplete Block Designs (BIBD). Two classes of BIBDs are used to create new construction methods for generating quasi- cyclic (QC) LDPC matrices. The method of constructing QC-LDPC codes using BIBDs introduced offer flexible matrices while preserving the QC structure whilst ensuring efficient and low complexity encoding using linear shift registers. The structure provides similar performance to that of MacKay codes over the Gaussian channel, whilst also showing very good performance over both the random and burst erasure channels. The construction of QC-LDPC codes using BIBDs methods also result in codes with fast decoding convergence and low error floors. The introduced QC-BIBD-LDPC codes are applied to the power-line environment, which inhibits both frequency-selective fading and severe noise characteristics. The effects of the frequency-selective fading of the channel are reduced-using a Multi-carrier- Modulation (MCM) technique known as Orthogonal Frequency Division Multiplexing (OFDM). This MCM improves the operating performance of the system by effectively creating a flat fading environment. With the presence of impulsive noise present in power-line communications, this work introduces a combination of recursive filtering in conjunction with>tht QC-BIBD-LDPC coding to mitigate its effects. To implement the sum-product algorithm (SPA) for decoding Low-Density Parity-Check (LDPC) codes in a power-line communications channel impaired by highly impulsive noise, it is essential to find the right channel probability distribution to optimize the algorithm. However, the computational complexity of the reliability factor is very high. This research proposes the use of a Kalman filter to help cancel out the degradation effect of impulsive noise and consequently approximate the output of the filter as a normal random variable. It is shown that LDPC-coded OFDM system in conjunction with Kalman filtering offers an efficient alternative solution to compensate for the disturbances caused by impulsive noise. Simulation results are provided to demonstrate the performance with the Kalman filter; it is shown that a gain of 15dB is achievable over a system equipped with a channellimiter. The performance evaluation of error-control codes for research and development purposes using software implementation can be a timely and complex process due to computational complex algorithms. Simulating codes to very low BitIBlock error rates is a computationally intense operation, but it is shown that many aspects of simulating LDPC codes are perfect for parallel computation. The speed and accuracy of the QC-LDPC simulations can be greatly increased by utilising the parallel architecture of graphics processing units (GPUs) from conventional "central processing" using CPUs. It is shown that simulating the QC-BIBD-LDPC codes in chapter 3 using Graphics Processing Units simulation speed-ups are achievable when compared to equivalent implementations using central processing units (CPUs). These speedups are related to an increase in simulation complexity, with speedups increasing with an increase in block-size, number of Sum- Product Algorithm (SPA) iterations and number of codewords decoded in parallel.

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.

Nonparametric tests have served as robust alternatives to traditional statistical tests with rigid underlying assumptions. If a researcher expects the treatment effects to follow an umbrella alternative, then the test developed in this research will be applicable in the Balanced Incomplete Block Design (Hemmer’s test). It is hypothesized that Hemmer’s test will prove to be more powerful than the Durbin test when the umbrella alternative is true. A mixed design consisting of a Balanced Incomplete Block Design and a Randomized Complete Block Design will also be considered, where two additional test statistics are developed for the umbrella alternative. Monte Carlo simulation studies were conducted using SAS to estimate powers. Various underlying distributions were used with 3, 4, and 5 treatments, and a variety of peaks and mean parameter values. For the mixed design, different ratios of complete to incomplete blocks were considered. Recommendations are given.

This thesis is based on four papers dealing with two different areas of mathematics.Paper I–III are in combinatorics, while Paper IV is in mathematical physics.In combinatorics, we work with design theory, one of whose applications aredesigning statistical experiments. Specifically, we are interested in symmetric incompleteblock designs (SBIBDs) and triple arrays and also the relationship betweenthese two types of designs.In Paper I, we investigate when a triple array can be balanced for intersectionwhich in the canonical case is equivalent to the inner design of the correspondingsymmetric balanced incomplete block design (SBIBD) being balanced. For this we derivenew existence criteria, and in particular we prove that the residual designof the related SBIBD must be quasi-symmetric, and give necessary and sufficientconditions on the intersection numbers. We also address the question of whenthe inner design is balanced with respect to every block of the SBIBD. We showthat such SBIBDs must possess the quasi-3 property, and we answer the existencequestion for all know classes of these designs.As triple arrays balanced for intersections seem to be very rare, it is natural toask if there are any other families of row-column designs with this property. In PaperII we give necessary and sufficient conditions for balanced grids to be balancedfor intersection and prove that all designs in an infinite family of binary pseudo-Youden designs are balanced for intersection.Existence of triple arrays is an open question. There is one construction of aninfinite, but special family called Paley triple arrays, and one general method forwhich one of the steps is unproved. In Paper III we investigate a third constructionmethod starting from Youden squares. This method was suggested in the literaturea long time ago, but was proven not to work by a counterexample. We show interalia that Youden squares from projective planes can never give a triple array bythis method, but that for every triple array corresponding to a biplane, there is asuitable Youden square for which the method works. Also, we construct the familyof Paley triple arrays by this method.In mathematical physics we work with solitons, which in nature can be seen asself-reinforcing waves acting like particles, and in mathematics as solutions of certainnon-linear differential equations. In Paper IV we study the non-commutativeversion of the two-dimensional Toda lattice for which we construct a family ofsolutions, and derive explicit solution formulas.
Denna avhandling baseras på fyra artiklar som behandlar två olika områden avmatematiken. Artikel I-III ligger inom kombinatoriken medan artikel IV behandlarmatematisk fysik.Inom kombinatoriken arbetar vi med designteori som bland annat har tillämpningardå man ska utforma statistiska experiment.I artikel I undersöker vi när en triple array kan vara snittbalanserad vilket i detkanoniska fallet är ekvivalent med den inre designen till den korresponderandesymmetriska balanserade inkompletta blockdesignen (SBIBD) är balanserad. För dettapresenterar vi nya nödvändiga villkor. Speciellt visar vi att den residuala designentill den korresponderande SBIBDen måste vara kvasi-symmetrisk och ger nödvändigaoch tillräckliga villkor för dess blockskärningstal. Vi adresserar ocksåfrågan om när den inre designen är balanserad med avseende på alla SBIBDensblock. Vi visar att en sådan SBIBD måste ha den egenskap som kallas kvasi-3 ochsvarar på existensfrågan för alla kända klasser av sådana designer.Eftersom snittbalanserade triple arrays verkar vara väldigt sällsynta är detnaturligt att fråga om det finns andra familjer av rad-kolumn designer som hardenna egenskap. I artikel II ger vi nödvändiga och tillräckliga villkor för att enbalanced grid ska vara snittbalanserad och visar att alla designer i en oändlig familjav binära pseudo-Youden squares är snittbalanserade.Existensfrågan för triple arrays är öppen fråga. Det finns en konstruktionsmetodför en oändlig men speciell familj kallad Paley triple arrays och så finns det enallmän metod för vilken ett steg är obevisat. I artikel III undersöker vi en tredjekonstruktionsmetod som utgår från Youden squares. Denna metod föreslogs i litteraturenför länge sedan men blev motbevisad med hjälp av ett motexempel. Vivisar bland annat att Youden squares från projektiva plan aldrig kan ge en triplearray med denna metod, men att det för varje triple array som korresponderartill ett biplan, så finns det en lämplig Youden square för vilken metoden fungerar.Vidare konstruerar vi familjen av Paley triple arrays med denna metod.Inom matematisk fysik arbetar vi med solitoner som man i naturen kan få sesom självförstärkande vågor vilka beter sig som partiklar. Inom matematiken ärde lösningar till vissa ickelinjära differentialekvationer. I artikel IV studerar vi dettvådimensionella Toda-gittret för vilken vi konstruerar en familj av lösningar ochäven explicita lösningsformler.

We study a family of random walks defined on certain Euclidean lattices that are related to incidence matrices of balanced incomplete block designs. We estimate the return probability of these random walks and use it to determine the asymptotics of the number of balanced incomplete block design matrices. We also consider the problem of collisions of independent simple random walks on graphs. We prove some new results in the collision problem, improve some existing ones, and provide counterexamples to illustrate the complexity of the problem.

The purpose of this paper is to present a new nonparametric test statistic for testing against ordered alternatives in a Balanced Incomplete Block Design (BIBD). This test will then be compared with the Durbin test which tests for differences between treatments in a BIBD but without regard to order. For the comparison, Monte Carlo simulations were used to generate the BIBD. Random samples were simulated from: Normal Distribution; Exponential Distribution; T distribution with three degrees of freedom. The number of treatments considered was three, four and five with all the possible combinations necessary for a BIBD. Small sample sizes were 20 or less and large sample sizes were 30 or more. The powers and alpha values were then estimated after 10,000 repetitions.The results of the study show that the new test proposed is more powerful than the Durbin test. Regardless of the distribution, sample size or number of treatments, the new test tended to have higher powers than the Durbin test.

Design optimality for treatment comparison experiments has been intensively studied by numerous researchers, employing a variety of statistically sound criteria. Their general formulation is based on the idea that optimality functions of the treatment information matrix are invariant to treatment permutation. This implies equal interest in all treatments. In practice, however, there are many experiments where not all treatments are equally important. When selecting a design for such an experiment, it would be better to weight the information gathered on different treatments according to their relative importance and/or interest. This dissertation develops a general theory of weighted design optimality, with special attention to the block design problem. Among others, this study develops and justifies weighted versions of the popular A, E and MV optimality criteria. These are based on the weighted information matrix, also introduced here. Sufficient conditions are derived for block designs to be weighted A, E and MV-optimal for situations where treatments fall into two groups according to two distinct levels of interest, these being important special cases of the "2-weight optimality" problem. Particularly, optimal designs are developed for experiments where one of the treatments is a control. The concept of efficiency balance is also studied in this dissertation. One view of efficiency balance and its generalizations is that unequal treatment replications are chosen to reflect unequal treatment interest. It is revealed that efficiency balance is closely related to the weighted-E approach to design selection. Functions of the canonical efficiency factors may be interpreted as weighted optimality criteria for comparison of designs with the same replication numbers.
Ph. D.

This research study proposes a solution to deal with missing observations which is a common problem in real world datasets. A nonparametric approach is used because of its ease of use relative to the parametric approach that beleaguer the user with firm assumptions. The study assumes data is in an Incomplete Block (IBD) and Completely Randomized (CRD) Mixed Design. The scope of this research was limited to three, four and five treatments. Mersenne - Twister (2014) simulations were used to vary the design and to estimate the test statistic powers. Two test statistics are proposed if the user expects a non – decreasing order of differences in treatment means. They are both applicable in the cited mixed design. The tests combine Alvo and Cabilio (1995) and Jonckheere – Terpstra ((Jonckheere (1954), Terpstra (1952)) in two ways: standardizing the sum of the standardized statistics and standardizing the sum of the unstandardized statistics. Results showed that the former is better. Three tests are proposed for the umbrella alternative. The first, Mungai’s test, is only applicable in an IBD. The other two tests combine Mungai’s and Mack – Wolfe (1981) using the same methods described in the previous paragraph. The same conclusion holds except when the size of the IBD’s sample was equal to or greater than a quarter that of the CRD.

Messinstrumente stellen in der wissenschaftlichen Forschung ein wesentliches Element zur Erkenntnisgewinnung dar. Das Besondere an Messinstrumenten im Large-scale Assessment in der Bildungsforschung ist, dass diese normalerweise für jede Studie neu konstruiert werden und dass die Testteilnehmer verschiedene Versionen des Tests bekommen. Hierbei ergeben sich potentielle Gefahren für die Akkuratheit und Validität der Messung. Um solche Gefahren zu minimieren, sollten (a) die Ursachen für Verzerrungen der Messung und (b) mögliche Strategien zur Optimierung der Messinstrumente eruiert werden. Deshalb wird in der vorliegenden Dissertation spezifischen Fragestellungen im Rahmen dieser beiden Forschungsanliegen nachgegangen.
Measurement instruments are essential elements in the acquisition of knowledge in scientific research. Special features of measurement instruments in large-scale assessments of student achievement are their frequent reconstruction and the usage of different test versions. Here, threats for the accuracy and validity of the measurement may emerge. To minimize such threats, (a) sources for potential bias of measurement and (b) strategies to optimize measuring instruments should be explored. Therefore, the present dissertation investigates several specific topics within these two research areas.

碩士
南台科技大學
資訊管理系
93
Since internet is an open environment, everyone can access or communicate by network in everywhere. So, when internet becomes popular, everyone will concern about the questions of information security. This thesis will study and analyze two group-oriented cryptographic systems: conference key distribution system and multi-proxy signature scheme, respectively. In 2002, Chung, Choi, Kim and Lee proposed an efficient conference key distribution system, which is generated by employing symmetric balanced incomplete block design. In Chung et al. protocol, the symmetric balanced incomplete block design can reduce the message overhead to in the case of , but it requires two rounds of message exchange. In 2004, Lee and Hwang further proposed an improved conference key distribution system. They also showed that Chung et al. protocol suffered several security problems, and proposed an improved protocol. Unfortunately, both Chung et al. protocol and Lee-Hwang protocol are insecure. This thesis will show that both Chung et al. protocol and Lee-Hwang protocol are vulnerable to the malicious participant conspiracy attack, and then propose an improved protocol. For multi-proxy signature scheme, Lin-Wu-Hwang proposed a multi-proxy signature scheme for partial delegation with cheater identification in 2002. This thesis finds that Lin-Wu-Hwang scheme requires message overhead (where is the number of proxy signer) and one round of message exchange. Thus, this thesis will propose a multi-proxy signature scheme employing symmetric balanced incomplete block design, and reduces the message overhead to in the case of . During the multi-proxy signature generation phase, it only requires two rounds of message exchange. The performance is very important for group-orientated cryptographic systems. Thus, this thesis reduces the message overhead employing symmetric balanced incomplete block design. Two improved protocol are more efficient than Chung et al. protocol and Lin-Wu-Hwang improved protocol.

碩士
國立臺北大學
通訊工程研究所
95
Digital watermarking is a kind of technique by which information can be embedded into digital contents. In recent years, digital contents are extensively used, especially over the Internet. Digital copyright protection thus becomes a more and more serious problem. Digital watermarking may be a good solution due to its robustness and invisibility. This study aims to present a new watermarking approach base on the balanced incomplete block designs (BIBD). The watermark is embedded into the high frequency wavelet coefficients of an original image. Characteristics of human visual system are used to control the watermark strength. At the receiver site, the original image is used to extract the embedded watermark, and then the mathematical structure of the BIBD is used to identify the copyright date. Robustness is achieved by utilizing the mathematical structure of the BIBD. Because of the invisibility and big capacity of information are both satisfy, this approach is adaptable to cover communication.

Balanced Incomplete Block Designs and Binary Linear Codes are two combinatorial designs. Due to the vast application of codes in communication the field of coding theory progressed more rapidly than many other fields of combinatorial designs. On the other hand, Block Designs are applicable in statistics and designing experiments in different fields, such as biology, medicine, and agriculture. Finding the relationship between instances of these two designs can be useful in constructing instances of one from the other. Applying the properties of codes to corresponding instances of Balanced Incomplete Block Designs has been used previously to show the non-existence of some designs. In this research the relationship between (16,6,3)-designs and (25,12) codes was determined.

Dissertação de Mestrado em Estatística, Matemática e Computação apresentada à Universidade Aberta
Neste trabalho, após uma introdução geral ao Planeamento de Experiências serão abordados os modelos de Planos em Blocos em geral e em particular os Planos em Blocos Incompletos Equilibrados com Repetições. Esta temática para além de oferecer muitas perspectivas no âmbito das aplicações práticas, no âmbito científico revela-se muito prometedora dado apresentar ainda muitos pontos em aberto, tanto no que respeita à dedução dos modelos gerais de análise, como na construção dos planos,estudo das possíveis estruturas e combinações de parâmetros. A ocorrência da repetição de blocos é desejável tanto do ponto de vista económico como da facilidade de aplicação prática. Para além disso, nos casos problemáticos, em que por acidente se percam unidades experimentais, a existência de blocos repetidos revela-se de primordial importância. Este trabalho tem como principais objectivos abordar a análise estatística de um BIBD e de um BIBD com repetição de blocos (BIBDR) apresentando exemplos, assim como explorar algumas representações geométricas de planos com blocos.
In this work, after a general introduction to the Experimental Design the models of Blocks Designs will be approached in general and specifically the Balanced Incomplete Blocks Designs with repetitions. Besides to offer many perspectives in the context of the practical applications, in the scientific extent this theme turns out to be very promising when to present still many points in opened, so much in what it respects to the deduction of the general models of analysis was given, as in the construction of the plans, study of the possible structures and combinations of parameters. The incident of the repetition of blocks is desirable both from the economical point of view and of the easiness of practical application. For besides, in the problematic cases, in which by accident experimental unities are lost, the existence of repeated blocks is revealed of primordial importance. This work has like principal objectives board the statistical analysis of a BIBD and of a BIBD with repetition of blocks (BIBDR) presenting examples, as well as to explore some geometrical representations of plans with blocks.

The most significant challenge for next wireless generation is to work opportunistically on the spectrum without a fixed spectrum allocation. Cognitive Radio (CR) is the candidate technology to utilize spectrum white space, which requires the CR to change its operating channel as the white space moves. In a CR ad-hoc network, each node could tune to a different channel; as a result, it cannot communicate with other nodes. This different tuning is due to the difficulty of maintaining Common Control Channel (CCC) in opportunistic spectrum network, and keeping the nodes synchronized in ad-hoc network. The CR ad-hoc network requires a protocol to match tuning channels between ad-hoc nodes, namely, rendezvous channels. In this thesis, two distributed Medium Access Control (MAC) protocols are designed that provide proper rendezvous channel without CCC or synchronization. The Balanced Incomplete Block Design (BIBD) is used in both protocols to provide our protocols a method of rendezvous between CR ad-hoc nodes. In fact, the BIBD guarantees there is at least one common element between any two blocks. If the channels are assigned to the BIBD elements and the searching sequence to the BIBD block, there is a guarantee of a rendezvous at least in one channel for each searching sequence. The first protocol uses a single-BIBD sequence and a multi-channel sensing. Alternatively, the second protocol uses a multi-BIBD sequence and a single-channel sensing. The single-sequence protocol analysis is based on the discrete Markov Chain. At the same time, the sequence structure of the BIBD in a multi-sequence protocol is used to define the Maximum Time to Rendezvous (MTTR). The simulation results confirm that the protocols outperform other existing protocols with respect to Time to Rendezvous (TTR), channel utilization, and network throughput. In addition, both protocols fairly distribute the network load on channels, and share the channels fairly among network nodes. This thesis provides straight forward and efficiently distributed MAC protocols for the CR ad-hoc networks.

碩士
國立中央大學
工業管理研究所
94
In many situations, observations in factorial design experiments may be dependent. In this article, we provide a explicit method to construct balanced designs for 2n and 2n-1 factorial designs when observations in blocks are correlated with positive correlation. We first characterized the balanced design, and then illustrate the method through balanced 23 full factorial designs and 24-1 half replicated factorial designs , finally we explain why the design made by the method is a balanced design and provide an analytical proof of balanced designs for both 2n full factorial designs and 2n-1 half replicated factorial designs .

An incomplete Latin square is a v x v array with an empty n x n subarray with every row and every column containing each symbol at most once and no row or column with an empty cell containing one of the last n symbols. A set of t incomplete mutually orthogonal Latin squares of order v and hole size n is a set of t incomplete Latin squares (containing the same empty subarray on the same set of symbols) with a natural extension to the condition of orthogonality. The existence of such sets have been previously explored only for small values of t. We determine an asymptotic result for the existence of t incomplete mutually orthogonal Latin squares for general t requiring large holes, which we develop from our results on incomplete pairwise balanced designs and incomplete group divisible designs.
Graduate

Background: Resampling methods such as the Monte Carlo (MC) and Bootstrap Approach (BA) are very flexible tools for statistical inference. They are used in general in experiments with small sample size or where the parametric test assumptions are not met. They are also used in situations where expressions for properties of complex estimators are statistically intractable. However, the MC and BA methods require relatively large random samples to estimate the parameters of the full permutation (FP) or exact distribution. Objective: The objective of this research study was to develop an efficient statistical computational resampling method that compares two population parameters, using a balanced and controlled sampling design. The application of the new method, the balanced randomization (BR) method, is discussed using microarray data where sample sizes are generally small. Methods: Multiple datasets were simulated from real data to compare the accuracy and efficiency of the methods (BR, MC, and BA). Datasets, probability distributions, parameters, and sample sizes were varied in the simulation. The correlation between the exact p-value and the p-values generated by simulation provide a measure of accuracy/consistency to compare methods. Sensitivity, specificity, power function, false negative and positive rates using graphical and multivariate analyses were used to compare methods. Results and Discussions: The correlation between the exact p-value and those estimated from simulation are higher for BR and MC, (increasing somewhat with increasing sample size), much less for BA, and most pronounced for skewed distributions (lognormal, exponential). Furthermore, the relative proportion of 95%/99% CI containing the true p-value for BR vs. MC=3%/1.3% (p<0.0001) and BR vs. BA=20%/15% (p<0.0001). The sensitivity, specificity and power function of the BR method were shown to have a slight advantage compared to those of MC and BA in most situations. As an example, the BR method was applied to a microarray study to discuss significantly differentially expressed genes.
acase@tulane.edu

Dissertação de Mestrado em Estatística, Matemática e Computação apresentada à Universidade Aberta
O Delineamento Experimental em Blocos Incompletos reveste-se de uma enorme importância uma vez que faz a ponte entre a matemática aplicada e as aplicações da estatística em áreas tão diversas como a agricultura, a medicina, a biometria, a criptografia, a genética, a indústria, as ciências da educação, entre muitas outras. Tem por objetivo principal a obtenção da maior quantidade de informação possível, a partir de uma pesquisa experimental. Para tal, procede-se a uma análise comparativa entre as diferentes variedades ou tratamentos por forma a se poderem controlar as fontes de variação aleatórias através da divisão das unidades experimentais em blocos. Seguidamente, procede-se à síntese de cada um dos capítulos: No primeiro capítulo deste trabalho é feita uma introdução ao delineamento experimental em blocos. É também exposta a motivação para a abordagem desta temática. No segundo capítulo apresenta-se uma introdução histórica ao delineamento experimental em blocos. É também desenvolvido com algum detalhe o estudo dos planos em blocos incompletos equilibrados. No terceiro capítulo procede-se à investigação de casos particulares, nomeadamente dos planos com blocos repetidos, planos com diferentes dimensões e planos com número de réplicas variável, passando de seguida à classificação em famílias dos planos com repetições (BIBDR) - Balanced Incomplete Block Designs with Repeated Blocks (Planos em Blocos Incompletos Equilibrados com Repetições). A construção dos (BIBD) - Balanced Incomplete Block Designs com Repetições (BIBDR) é abordada como um Plano Otimal que proporciona facilidade de aplicação prática e se reveste de importância sob o ponto de vista económico, (Foody, W. & Hedayat, A, 1977), (Hedayat, A. S. & Hwang, H. L. 1984). Foi desenvolvido um programa em Basic para obtenção da lista de parâmetros dos possíveis BIBDR, com dimensão de bloco sete, tendo em conta as condições necessárias de construção dos BIBDR sobre certas restrições. Os planos são classificados atendendo às três famílias de BIBDR, definidas em (Hedayat, A. S. & Hwang, H. L. 1984). O capítulo quatro explora as ligações existentes entre desenho experimental, matrizes de Hadamard e o risco de perda de dados nos Códigos QR. No capítulo cinco aprofunda-se o estudo dos Códigos QR, explorando diversas aplicações e apresentando exemplos práticos. No capítulo seis expõe-se uma introdução ao software estatístico R (Project for Statistical Computing) e exemplos práticos desenvolvidos neste, tendo por base a temática do delineamento experimental com blocos. O capítulo sete apresenta diversos trabalhos de investigação com abordagens recentes ao delineamento experimental em blocos. Estes trabalhos abrangem temáticas diversas como a medicina, criptografia, ensaios clínicos, Códigos QR entre outros. Neste capítulo, destaca-se um estudo sobre a doença de Parkinson no âmbito dos ensaios clínicos. A aplicação prática presente no capítulo oito mostra a análise de fiabilidade inter-examinador inspirada nesse estudo. Nesta aplicação serão analisados seis neurologistas que examinam dez pacientes com Parkinson. No capítulo nove são sintetizadas algumas considerações finais e apresentadas perspetivas de investigação futura.
The Design of Experiments considering incomplete blocks is of great importance since it bridges the gap between Applied Mathematics and Applications of Statistics in diverse areas such as agriculture, medicine, biometrics, encryption, genetics, industry and science education, among many others. The Design of Experiments or Experimental Design has the main objective of obtaining as much information as possible from an experimental study. A comparative analysis of the different varieties or treatments is made, in order to be able to control the random sources of variation by dividing the experimental units in blocks. Subsequently, we will synthesize each one of the chapters: In the first chapter of this paper an introduction to Experimental Design in blocks is presented. It is also exposed the motivation to address this issue. The second chapter presents a historical introduction to Experimental Design in blocks. It is also approached the study of balanced incomplete block designs and respective characteristics. In the third chapter a research of special cases is developed, such as designs with repeated blocks, designs with different dimensions and designs with different number of replicas, followed by the classification in families of BIB designs with repetitions. The construction of BIBD with Repetitions (BIBDR) is carried out using Optimal Designs which provide ease of practical application and is of great importance from the economical point of view, (Foody, W. & Hedayat, A. 1977), (Hedayat, A. S. & Hwang, H. L.1984). A program in BASIC language has been developed in order to obtain the list parameter of possible BIBDR with seven block size, taking into account certain restrictions. These designs are classified according to the three families of BIBDR defined in (Hedayat, A. S. & Hwang, H. L.1984). Chapter four explores the links between Experimental Design, Hadamard matrices and the risk of data loss in QR Codes. Chapter five deepens the study of QR Codes, exploring different applications and presenting practical examples. Chapter six presents an introduction to statistical software R (Project for Statistical Computing) and practical examples developed in this, based on the theme of Experimental Design with blocks. The seventh chapter presents recent approaches and applications of Experimental Design considering blocks. These works cover different topics such as medicine, encryption, clinical tests, QR Codes and more. The approach of the previous section on clinical studies highlighted a study of Parkinson's disease. Chapter eight shows a virtual practical application of this, in the analysis of an inter reliability inspired in this study. In this application a design considering six neurologists who examined ten patients with Parkinson's disease will be considered. In chapter nine, some final remarks are summarized and perspectives for future research are presented.