Cyclic division algebras in the space-time coding

• Grant number: 11/12657-9
• Support Opportunities: Scholarships abroad - Research
• Start date: January 15, 2012
• End date: July 14, 2012
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Carina Alves
• Grantee: Carina Alves
• Host Investigator: Jean-Claude Belfiore
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil
• Institution abroad: ParisTech, France

Abstract

Division algebras have been proposed as a new tool for constructing Space-Time codes, since they are non-commutative algebras that naturally yield linear fully diverse codes. For transmission over quasi-static MIMO fading channels with n transmit antennas, diversity can be obtained by using an inner fully diverse space-time block code while coding gain, derived from the determinant criterion, comes from an appropriate outer code. When the inner code has a cyclic al gebra structure over a number field, as for perfect space-time codes, an outer code can be designed viacoset coding. More precisely, if we take the quotient of the algebra by a two-sided ideal this leads to a finite alphabet for the outer code, with a cyclic algebra structure over a finite field or a finite ring. We will establish a general framework for designing coset codes via a series of isomorphisms that allows to represent the outer code alphabet in three different ways: an algebra of matrices over a finite ring, a cyclic algebra over a finite ring, and the Cartesian product of finite rings. The decoding of algebraic space-time codes has been performed using their lattice point representation. Belfiore et al introduced a right preprocessing method for the decoding of space-time block codes based on quaternion algebras, which allows to improve the performance of suboptimal decoders and reduces the complexity of ML decoders. So another question related to cyclic division algebras is deal with the generalization of algebraic reduction to higher-dimensional space-time codes based on cyclic division algebras. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

ALVES, C.; BELFIORE, J. -C.. Lattices from maximal orders into quaternion algebras. Journal of Pure and Applied Algebra, v. 219, n. 4, p. 687-702, APR 2015. (11/12657-9)