Space-time block codes from nonassociative division algebras

Space-time coding is used for reliable high rate transmission over wireless digital channels with multiple antennas at both the transmitter and receiver ends. From a mathematical point of view, a space-time block code (STBC) consists of a family of matrices with complex entries (the codebook) that satisfies a number of properties which determine how well the code performs. The first aim is to find fully diverse codebooks, where the difference of any two code words has full rank. Once a fully diverse codebook is found it is then further optimized to satisfy additional design criteria (see Section 6). Using central simple associative division algebras to build space-time block codes allows for a systematic code design (see for instance [22], [13], [29], [15], [7], [8], [9] and the excellent survey [28]). Most of the existing codes are built from cyclic division algebras over F = Q(i) or F = Q(ζ3) with ζ3 = e 2πi/3 a third root of unity. These fields are used for the transmission of QAM or HEX constellations, respectively. There are two ways to embed an associative division algebra into a matrix algebra in order to obtain a codebook: the left regular representation of the algebra and the representation over some maximal subfield. For instance, a real 4 × 4 orthogonal design is obtained by the left regular representation of the real quaternions H = (−1,−1)R (see for instance [30, p. 1458]), whereas the Alamouti code [1] uses the representation of H over its maximal subfield C. In [29, p. 2608], the authors note that “the Alamouti code is the only rate-one STBC which is full rank over any finite subset of C, which is due to the fact that the set of quaternions H is the only division algebra which has the entire complexes