## 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 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.

Original language | English |
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Pages (from-to) | 6060-6082 |

Number of pages | 23 |

Journal | IEEE Transactions on Information Theory |

Volume | 59 |

Issue number | 9 |

DOIs | |

Publication status | Published - Sep 2013 |

MoE publication type | A1 Journal article-refereed |

## Keywords

- Algebra
- diversity-multiplexing gain tradeoff (DMT)
- division algebra
- Lie groups
- multiple-input multiple-output (MIMO)
- number theory
- space-time block codes (STBCs)
- unit group
- Zeta functions
- TIME BLOCK-CODES
- HOMOGENEOUS VARIETIES
- DIVERSITY TECHNIQUE
- SPACE
- ORDERS
- TRADEOFF
- LATTICES
- POINTS