Inverse Determinant Sums and Connections Between Fading Channel Information Theory and Algebra

• Published in: IEEE transactions on information theory Vol. 59; no. 9; pp. 6060 - 6082
• Main Authors: Vehkalahti, R., Hsiao-Feng Lu, Luzzi, L.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.09.2013; Institute of Electrical and Electronics Engineers
• Subjects: Algebra; Applied sciences; Asymptotic properties; Coding, codes; Computer Science; Determinants; diversity-multiplexing gain tradeoff (DMT); division algebra; Encoding; Error probability; Exact sciences and technology; Fading; Gain; Information Theory; Information, signal and communications theory; Inverse; Lattices; Lie groups; Mathematics; multiple-input multiple-output (MIMO); Multiplexing; Number Theory; Rings and Algebras; Signal and communications theory; Signal to noise ratio; space-time block codes (STBCs); Space-time codes; Sums; Systems, networks and services of telecommunications; Telecommunications; Telecommunications and information theory; Transmission and modulation (techniques and equipments); unit group; Zeta functions; Fading channels; Multiplexing; MIMO system; Ergodic theory; Block code; division algebra; unit group; Space-time codes; Error probability; diversity-multiplexing gain tradeoff (DMT); space-time block codes (STBCs); Algebraic code; multiple-input multiple-output (MIMO); Zeta functions; Inverse problem; Algebra; Lie groups; Lie group; Number theory; Information theory
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2013.2266396

This work considers inverse determinant sums, which arise from the union bound on the error probability, as a tool for designing and analyzing algebraic space-time block codes. A general framework to study these sums is established, and the connection between asymptotic growth of inverse determinant sums and the diversity-multiplexing gain tradeoff is investigated. It is proven that the growth of the inverse determinant sum of a division algebra-based space-time code is completely determined by the growth of the unit group. This reduces the inverse determinant sum analysis to studying certain asymptotic integrals in Lie groups. Using recent methods from ergodic theory, a complete classification of the inverse determinant sums of the most well-known algebraic space-time codes is provided. The approach reveals an interesting and tight relation between diversity-multiplexing gain tradeoff and point counting in Lie groups.