## Abstract

It is shown why the discriminant of a maximal order within a cyclic division algebra must be minimized in order to get the densest possible matrix lattices with a prescribed nonvanishing minimum determinant. Using results from class field theory, a lower bound to the minimum discriminant of a maximal order with a given center and index (= the number of Tx/Rx antennas) is derived. Also numerous examples of division algebras achieving the bound are given. For example, a matrix lattice with quadrature amplitude modulation (QAM) coefficients that has 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant is constructed. Also, a general algorithm due to Ivanyos and Ronyai for finding maximal orders within a cyclic division algebra is described and enhancements to this algorithm are discussed. Also some general methods for finding cyclic division algebras of a prescribed index achieving the lower bound are proposed.

Original language | English |
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Pages (from-to) | 3751-3780 |

Number of pages | 30 |

Journal | IEEE Transactions on Information Theory |

Volume | 55 |

Issue number | 8 |

DOIs | |

Publication status | Published - 2009 |

MoE publication type | A1 Journal article-refereed |

Event | IEEE International Symposium on Information Theory - Seattle, United States Duration: 9 Jul 2006 → 14 Jul 2006 |

## Keywords

- Cyclic division algebras (CDAs)
- dense lattices
- discriminants
- Hasse invariants
- maximal orders
- multiple-input multiple-output (MIMO) channels
- multiplexing
- space-time block codes (STBCs)
- SPACE-TIME CODES
- BLOCK-CODES
- MAXIMAL-ORDERS
- PERFECT SPACE
- DIVERSITY
- CONSTRUCTION
- PERFORMANCE
- TRADEOFF
- ANTENNAS
- NUMBER