Central Simple Algebras: an application to wireless communications

Introduction The aim of this work is to show how central simple algebras, and their arith- metics, can be a useful tool in modelling, encoding, decoding and detecting interesting proprieties of wireless communication. The basic concept is that the generic signal received through a wireless chan- nel can be expressed, in a matricial form, as Y =HX +V where Y is the received signal, X is the sent signal, V is the noise and H is the fading matrix. The main problem, in wireless communication, is given by the presence of the fading matrix, that is intrinsic in the nature of this kind of trasmissions. So it is necessary to send X in a particular form. For example we want that multiplication H X is not zero, assuming X;H6= 0, or that the determi- nants of two distinct matrices are dierent; these, and other, problems nd a natural resolution in theory of algebras. In fact it is possible to embed central simple algebras, in the matrices al- gebras and nd algebras that do not have zero-divisors and with non-zero determinant, the division algebras, using the results of this theory. Another problem arises from the fact that the received signals are points in the space of matrices, so we want some discrete structures that plays the same role of the nite-dimensional vector-spaces, over nite elds, in the standard error-correcting codes theory. To solve this problem we introduce orders and various concepts of algebraic number theory. We present also some methods for decoding; we do not concentrate on this 1