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 our bound are given. E.g. we construct a matrix lattice with QAM coefficients that has 2.5 times as many codewords as the celebrated Golden code of the same minimum determinant. We describe a general algorithm due to Ivanyos and Rónyai for finding maximal orders within a cyclic division algebra and discuss our enhancements to this algorithm. We also consider general methods for finding cyclic division algebras of a prescribed index achieving our lower bound. Index TermsCyclic division algebras, dense lattices, discriminants, Hasse invariants, maximal orders, multiple-input multipleoutput (MIMO) channels, multiplexing, space-time block codes (STBCs).I. OVERVIEW Multiple-antenna wireless communication promises very high data rates, in particular when we have perfect channel state information (CSI) available at the receiver. In [1] the design criteria for such systems were developed, and further on the evolution of space-time (ST) codes took two directions: trellis codes and block codes. Our work concentrates on the latter branch.We are interested in the coherent multiple input-multiple output (MIMO) case. A lattice is a discrete finitely generated free abelian subgroup L of a real or complex finite dimensional vector space V, called the ambient space. In the space-time setting a natural ambient space is the space M n (C) of complex n×n matrices. We only consider full rank lattices that have a basis x 1 , x 2 , . . . , x 2n 2 consisting of matrices that are linearly independent over the field of real numbers. We can form a 2n 2 × 2n 2 matrix M having rows consisting of the real and imaginary parts of all the basis elements. It is well known that the measure, or hypervolume, m(L) of the fundamental parallelotope of the lattice then equals the absolute value of det(M). Alternatively we may use the Gram matrix 2 . From the pairwise error probability (PEP) point of view [2], the performance of a space-time code is dependent on two parameters: diversity gain and coding gain. Diversity gain is the minimum of the rank of the difference matrix X − X ′ taken over all distinct code matrices X, X ′ ∈ C, also called the rank of the code C. When C is full-rank, the coding gain is proportional to the determinant of the matrix

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We construct explicit rate-one, full-diversity, geometrically dense matrix lattices with large, non-vanishing determinants (NVD) for four transmit antenna multiple-input single-output (MISO) space-time (ST) applications. The constructions are based on the theory of rings of algebraic integers and related subrings of the Hamiltonian quaternions and can be extended to a larger number of Tx antennas. The usage of ideals guarantees a non-vanishing determinant larger than one and an easy way to present the exact proofs for the minimum determinants. The idea of finding denser sublattices within a given division algebra is then generalized to a multiple-input multiple-output (MIMO) case with an arbitrary number of Tx antennas by using the theory of cyclic division algebras (CDA) and maximal orders. It is also shown that the explicit constructions in this paper all have a simple decoding method based on sphere decoding. Related to the decoding complexity, the notion of sensitivity is introduced, and experimental evidence indicating a connection between sensitivity, decoding complexity and performance is provided. Simulations in a quasi-static Rayleigh fading channel show that our dense quaternionic constructions outperform both the earlier rectangular lattices and the rotated ABBA lattice as well as the DAST lattice. We also show that our quaternionic lattice is better than the DAST lattice in terms of the diversity-multiplexing gain tradeoff.Comment: 24 pages, 17 figures, revised version submitted to IEEE Trans. on Inform. Theory Dec. 2007, first version submitted Sep. 200

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