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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.
- Hadamard matrices and links to information theoryPublication . Francisco, Carla; Oliveira, Teresa; Oliveira, Amilcar; Grilo, LuisThe existence of Hadamard matrices remains one of the most challenging open questions in combinatorics. Substantial progress on their existence has resulted from advances in algebraic design theory using deep connections with linear algebra, abstract algebra, finite geometry, number theory, combinatorics and optimization. The construction and analysis of Hadamard matrices, and their use on combinatorial designs, play an important role nowadays in diverse fields such as; quantum information, communications, networking, cryptography, biometry and security. Hadamard Matrices are present in our daily life and they give rise to a class of block designs named Hadamard configurations. Different applications of it based on new technologies and codes of figures such as QR Codes are present almost everywhere. BIBD are very well known as a tool to solve emerging problems in this area. Illustrations and applications to authentication codes and secret sharing schemes will be presented.