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- Hadamard matrices on error detection and correction: useful links to BIBDPublication . Francisco, Carla; Oliveira, Teresa A.; Oliveira, Amilcar; Carvalho, FranciscoIn the areas of Computer Science and Telecommunications there is a huge amount of applications in which error control, error detection and error correction are crucial tools to enable reliable delivery of digital data over unreliable communication, thus providing quality of service. Hadamard matrices can almost directly be used as an error-correcting code using an Hadamard code, generalized in Reed-Muller codes. Advances in algebraic design theory by using deep connections with algebra, finite geometry, number theory, combinatorics and optimization provided a substantial progress on exploring Hadamard matrices. Their construction and its use on combinatorics are crucial nowadays in diverse fields such as: quantum information, communications, networking, cryptography, biometry and security. Hadamard matrices give rise to a class of block designs named Hadamard configurations and different applications of it based on new technologies and codes of figures such as QR Codes are present almost everywhere. Some connections to Balanced Incomplete Block Designs are very well known as a tool to solve emerging problems in these areas. We will explore the use of Hadamard matrices on QR Codes error detection and correction. Some examples will be provided.
- Using R in experimental design with BIBD: an application in health sciencesPublication . Oliveira, Teresa; Francisco, Carla; Oliveira, Amilcar; Ferreira, AgostinhoConsidering the implementation of an Experimental Design, in any field, the experimenter must pay particular attention and look for the best strategies in the following steps: planning the design selection, conduct the experiments, collect observed data, proceed to analysis and interpretation of results. The focus is on providing both - a deep understanding of the problem under research and a powerful experimental process at a reduced cost. Mainly thanks to the possibility of allowing to separate variation sources, the importance of Experimental Design in Health Sciences is strongly recommended since long time. Particular attention has been devoted to Block Designs and more precisely to Balanced Incomplete Block Designs - in this case the relevance states from the fact that these designs allow testing simultaneously a number of treatments bigger than the block size. Our example refers to a possible study of inter reliability of the Parkinson disease, taking into account the UPDRS (Unified Parkinson’s disease rating scale) in order to test if there are significant differences between the specialists who evaluate the patients performances. Statistical studies on this disease were described for example in Richards et al (1994), where the authors investigate the inter-rater Reliability of the Unified Parkinson’s Disease Rating Scale Motor Examination. We consider a simulation of a practical situation in which the patients were observed by different specialists and the UPDRS on assessing the impact of Parkinson’s disease in patients was observed. Assigning treatments to the subjects following a particular BIBD(9,24,8,3,2) structure, we illustrate that BIB Designs can be used as a powerful tool to solve emerging problems in this area. Once a structure with repeated blocks allows to have some block contrasts with minimum variance, see Oliveira et al. (2006), the design with cardinality 12 was selected for the example. R software was used for computations.
- BIB designs with repeated blocks: review and perspectivesPublication . Oliveira, TeresaExperimental Design plays an important role on establishing an interface between Applied Mathematics and statistical applications in several fields, like Agriculture, Industry, Genetics,Biology and Education Sciences. The goal of any Experimental Design is to obtain the maximum amount of information for a given experimental effort, to allow comparisons between varieties and to control for sources of random variability. Randomized block designs are used to control for these sources. A Balanced Incomplete Block Design (BIB Design) is a randomized block design with number of varieties greater than the block size and with all pairs of varieties occurring equally often along the blocks. The Fisher related information of a balanced block design will remain invariant whether or not the design has repeated blocks. This fact can be used theoretically to build a large number of non-isomorphic designs for the same set of design parameters, which could be used for many different purposes both in experimentations and surveys from finite populations. The original and most important method on the construction of BIB Designs with repeated blocks (BIBDR) is due to Hedayat and Li (1979): the trade-off method. Since then, many authors and researchers have been paying particular attention to the construction of BIBDR, but still some unsolved problems remain. This issue will be briefly reviewed and new results on the existence and construction of BIBDR, as well as several unsolved problems for further research will be presented.