The Coding Gain of Real Matrix Lattices: Bounds and Existence Results

The paper considers the question of the normalized minimum determinant (or asymptotic coding gain) of real matrix lattices. The coding theoretic motivation for such study arises, for example, from the questions considering multiple-input multiple-output (MIMO) ultra-wideband (UWB) transmission. At the beginning, totally general coding gain bounds for real MIMO lattice codes is given by translating the problem into geometric language. Then code lattices that are produced from division algebras are considered. By applying methods from the theory of central simple algebras, coding gain bounds for code lattices coming from orders of division algebras are given. Finally, it is proven that these bounds can be reached by using maximal orders. In the case of 2 × 2 matrix lattices, this existence result proves that the general geometric bound derived earlier can be reached.