Inverse Determinant Sums and Connections between Fading Channel Information Theory and Algebra
Roope Vehkalahti, Hsiao-Feng Francis Lu, Senior Member, Laura Luzzi · 2013
Abstract—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 gen-eral framework to study these sums is established, and the con-nection 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 divi-sion algebra-based space-time code is completely determined by the growth of the unit group. This reduces the inverse determi-nant sum analysis to studying certain asymptotic integrals in Lie groups. Using recentmethods from ergodic theory, a complete clas-sification 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. Index Terms—Algebra, diversity-multiplexing gain tradeoff (DMT), division algebra, Lie groups, multiple-input mul-tiple-output (MIMO), number theory, space-time block codes