Maximal orders in space-time coding: Construction and decoding

Previously, it was 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 non-vanishing minimum determinant. In this paper, the actual procedure of constructing maximal orders is described in more detail, aiming to provide a handy tool also for researchers with only a modest mathematical background. For instance, it is explicitly shown, step by step, how to construct a matrix lattice with QAM coefficients that has 2.5 times as many codewords as the famous Golden code of the same minimum determinant. In order to decode maximal order based space-time codes, the usual sphere decoder has to be modified. A pseudo algorithm describing the additional steps is given. For the algorithm to function it is essential that we also speed up the search for the shortest lattice vectors ensuring in this way that the usage of a codebook becomes feasible. Both the search and the decoding can be performed by adding an upper bound on the energy of the single vector in use.