BIBD, Hadamard matrices and new technological devices: Applications to QR codes

Abstract

Experimental design utilizing blocks was first presented by Ronald A. Fisher in the beginning of the 20th century with applications in agriculture. It became one of the areas of mass development for research in various fields, from agriculture to medicine, playing a central role not only on the research process but also on the new technological advances. As a branch of block designs, research on Balanced Incomplete Block Designs (BIBD) arose several interesting and defying problems within combinatory mathematics. BIBD designs can be constructed and analysed more easily with the aid of computer tools, such as the The R Project for Statistical Computing, which, as will be shown, contains several different tools just for this purpose. Hadamard matrices are present in our daily life, they give rise to a class of block designs named Hadamard configurations. It is easy and current to find their different applications based on new technologies and codes of figures such as Quick Response Codes (QR Codes). These are bi-dimensional bar codes that can be easily read by common devices which have image capture function, such as mobile phones. The use of such codes is very popular nowadays in simple things like to send a text message, a picture or for a batch of information regarding a product or whatever one desires. Error correction on these QR Codes make use of Hadamard matrices. The importance of using Qr Codes in cryptocurrencies and their links between the various methodologies using R software will be explored, as well as connections with Hadamard Matrices and BIBD.