Space-Time Codes

In 1998 Tarokh, Seshadri, and Calderbank [1] introduced the space-time codes as a solution to protect data against errors in fading channels of mobile networks. Space-time codes can be used in multiple-input multiple-output (MIMO) channels where several transmitting and receiving antennas are used. The very first explicit space-time code for two transmitting antennas was introduced by Alamouti in [2].

The task of constructing good space-time codes is closely related to constructing set of complex matrices that have big minimum determinant. Sethuraman, Rajan, and Shashidhar [3] showed how good matrix lattices can be constructed with the aid of division algebras. Since then, the search for good space-time codes from division algebras has been of the utmost interest to researchers in coding theory, number theory, and algebra, one of the breakthroughs being by no doubt the discovery of the so-called Perfect codes by Oggier et al. [4]

The main contribution of our group to space-time coding is the introduction of maximal orders of cyclic division algebras in the design of dense space-time codes [6], [7]. The concept of maximal orders is not new, see e.g. the excellent book "Maximal Orders" by Irving Reiner from 1975. The theory of maximal orders in division algebras and more generally in central simple algebras allows us to construct several families of record breaking space-time codes.

References:
1. Vahid Tarokh, Nambi Seshadri, and A. R. Calderbank, Space-time codes for high rate wireless communication: performance criterion and code construction, IEEE Transactions on Information Theory, 44(2):744-765, 1998.
2. S. M. Alamouti, A simple transmit diversity technique for wireless communication, IEEE Journal on Selected Areas in Communications, 16(8):1451-1458, 1998.
3. B. A. Sethuraman, B. Sundar Rajan, and V. Shashidhar, Full-diversity, high-rate space-time block codes from division algebras, IEEE Transactions on Information Theory, 49(10):2596-2616, 2003.
4. F. E. Oggier, G. Rekaya, J.-C. Belfiore, E. Viterbo. Perfect Space Time Block Codes, IEEE Transactions on Information Theory, 52(9):3885-3902, 2006.
5. J. Hiltunen, J. Lahtonen, and A. Nikkanen, On the BCJR-like trellis for space-time group codes, Proceedings of the 2004 International Symposium on Information Theory, page 158. [doi]
6. C. Hollanti, J. Lahtonen, and H.-F. Lu, Maximal orders in the design of dense space-time lattice codes, IEEE Transactions on Information Theory, 54(10):4493--4510, 2008.
7. R. Vehkalahti, C. Hollanti, J. Lahtonen, and K. Ranto, On the densest MIMO lattices from cyclic division algebras, IEEE Trans. on Inform. Theory, 55(8):3751-3780, 2009. Available from ArXiv.