Below we will prove the nonexistence of full NVD multiuser codes when each user is equipped with single transmit antenna. Thus, as there are two users in the present case, the overall code matrix is of size (2 × 2), one row for each user. In the following we show if the (2 × 2) code matrix has non-zero determinant then it cannot have NVD. We first invoke the following well-known result in lattice theory.
Lemma 31. A subgroup in Cnis a lattice if and only if it is discrete.
To prove Theorem 26, let us suppose that user one uses a code C1 that is a full lattice, i.e. it has 4 generators as an abelian group in C2. The reason for having 4 generators is that the transmission of code takes two channel uses, and in each channel use it is a complex baseband symbol that has two components, the in-phase and quadrature. Let us now suppose that (b1, b2) is some non-zero codeword sent by the second user and (a1, a2) a nonzero codeword sent by the first user. The two-user matrix is now
S = a1 a2 b1 b2
.
We have det(S) = a1b2 − a2b1. Fixing (b1, b2) for the second user gives us an idea of a natural homomorphism f from C1 to C where (x1, x2) 7→ x1b2 − x2b1. The assumption of S having non-zero determinant for all nonzero (a1, a2) ∈ C1 suggests that x1b2− x2b1 is zero if and only if (x1, x2) is zero, hence we see that f is a group isomorphism from C1 to f (C1) ⊆ C. Now f (C1) is a subgroup in C and it must have 4 generators as an abelian group because it is isomorphic to C1. As any lattice in C can have at maximum 2 generators, we see that f (C1) cannot be a lattice. Therefore it must have an accumulation point. Because f (C1) is a group we can suppose that it has an accumulation point at 0. This means that there exists an element (a1, a2) in C1 so that we can get |a1b2 − a2b1| arbitrarily small, yielding a vanishing determinant. Hence this proves there does not exist any multiuser codes that are full NVD.
Bibliography
[1] F. Oggier, G. Rekaya, J.-C. Belfiore, and E. Viterbo, “Perfect space time block codes,”
IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3885–3902, Sept. 2006.
[2] P. Elia, B. A. Sethuraman, and P. V. Kumar, “Perfect space-time codes for any number of antennas,” IEEE Trans. Inf. Theory, vol. 53, no. 11, pp. 3853–3868, 2007.
[3] H.-F. Lu, “Explicit constructions of multi-block space-time codes that achieve the diversity-multiplexing tradeoff,” in Proc. 2006 IEEE Int. Symp. Inform. Theory, Seat-tle, WA, Jul. 2006, pp. 1149–1153.
[4] K. R. Kumar and G. Caire, “Space-time codes from structured lat-tices,” IEEE Trans. Inf. Theory, to appear, 2008. Preprint available at http://www.citebase.org/abstract?id=oai:arXiv.org:0804.1811.
[5] R. Vehkalahti, C. Hollanti, J. Lahtonen, and K. Ranto, “On the densest MIMO lat-tices from cyclic division algebras,” IEEE Trans. on Inform. Theory, to appear, 2009.
Preprint available at: http://arxiv.org/abs/cs.IT/0703052.
[6] “Multiuser space-time/frequency code design,” in Proc. 2006 IEEE Int. Symp. Inform.
Theory, Seattle, WA, Jul. 2006, pp. 2819 – 2823.
[7] D. Tse, P. Viswanath, and L. Zheng, “Diversity and multiplexing tradeoff in multiple-access channels,” IEEE Trans. Inf. Theory, vol. 50, no. 9, pp. 1859–1874, 2004.
[8] L. Zheng and D. Tse, “Diversity and multiplexing: a fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003.
[9] Y. Nam and H. E. Gamal, “On the optimality of lattice coding and decoding in mul-tiple access channels,” in Proc. 2007 IEEE Int. Symp. Inform. Theory, Nice, France, Jun. 2007.
[10] Y. Hong and E. Viterbo, “Algebraic multi-user space-time block codes for 2x2 MIMO,” in Proc. 2008 IEEE PIMRC, Cannes, France, Sep. 2008.
[11] M. Badr and J.-C. Belfiore, “Distributed space-time block codes for the MIMO mul-tiple access channel,” in Proc. 2008 IEEE Int. Symp. Inform. Theory, Toronto, ON, Jul. 2008.
[12] W. Zhang and K. B. Letaief, “A systematic design of multiuser space-frequency codes for MIMO-OFDM systems,” in Proc. 2007 IEEE Int. Conf. Commun., Jul. 2007, pp.
1054–1058.
[13] P. Coronel, M. G¨artner, and H. B¨olcskei, “Diversity multiplexing tradeoff in selec-tive fading multiple-access MIMO channels,” in Proc. 2008 IEEE Int. Symp. Inform.
Theory, Toronto, ON, Jul. 2008.
[14] H.-F. Lu, C. Hollanti, R. Vehkalahti, and J. Lahtonen, “DMT optimal code construc-tions for multiuser MIMO channel,” submitted to IEEE Trans. Inf. Theory, April 2009.
[15] F. E. Oggier, J.-C. Belfiore, and E. Viterbo, “Cyclic division algebras: A tool for space-time coding,” Foundations and Trends in Communications and Information Theory, vol. 4, no. 1, pp. 1–95, 2007.
[16] I. Reiner, Maximal Orders. New York: Academic Press, 1975.
[17] P. Elia and P. V. Kumar, “Approximately-universal space-time codes for the paral-lel, multi-block and cooperative-dynamic-decode-and-forward channels,” Jul. 2007, http://arxiv.org/pdf/0706.3502.
[18] C. Hollanti and J. Lahtonen, “A new tool: Constructing STBCs from maximal orders in central simple algebras,” in Proc. 2006 IEEE Inform. Theory Workshop, Punta del Este, Uruguay, Mar. 13-17 2006.
[19] J. Liu and A. R. Calderbank, “The icosian code and the e8 lattice: A new 4 × 4 space-time code with nonvanishing determinant,” in Proc. 2006 IEEE Int. Symp. Inform.
Theory, Seattle, WA, Jul. 2006, pp. 1006–1010.
[20] C. Hollanti, J. Lahtonen, and H.-F. Lu, “Maximal orders in the design of dense space-time lattice codes,” IEEE Trans. Inf. Theory, vol. 54, no. 10, pp. 4493 – 4510, Oct.
2008.
[21] H. Lu, J. Lahtonen, R. Vehkalahti, and C. Hollanti, “Remarks on the criteria of con-structing MIMO-MAC DMT optimal codes,” http://arxiv.org/abs/0908.3166.
[22] S. Yang and J.-C. Belfiore, “Optimal space-time codes for the amplify-and-forward cooperative channel,” in Proc. 43nd Allerton Conference on Communication, Control and Computing, Monticello, Illinois, Sep. 2005.
[23] H.-F. Lu, “Explicit constructions of multi-block space-time codes that achieve the diversity-multiplexing tradeoff,” IEEE Trans. Inf. Theory, vol. 54, no. 8, pp. 3790–
3796, 2008.
[24] P. Elia, K. R. Kumar, S. A. Pawar, P. V. Kumar, and H.-F. Lu, “Explicit construction of space-time block codes achieving the diversity-multiplexing gain tradeoff,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3869–3884, Sep. 2006.
[25] T. Kiran and B. S. Rajan, “STBC-schemes with non-vanishing determinant for certain number of transmit antennas,” IEEE Trans. Inf. Theory, vol. 51, no. 8, pp. 2984–2992, Aug. 2005.
[26] J.-C. Belfiore, G. Rekaya, and E.Viterbo, “The Golden code: a 2 × 2 full-rate space-time code with non-vanishing determinants,” IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1432–1436, Apr. 2005.
[27] C. K¨ose and R. D. Wesel, “Universal space-time trellis codes,” IEEE Trans. Inf. The-ory, vol. 49, no. 10, pp. 2717–2727, Oct. 2003.
[28] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge: Cambridge University Press, 1985.
[29] C. Hollanti and K. Ranto, “Maximal orders in space-time coding: Construction and decoding,” in Proc. 2008 Int. Symp. Inf. Theory and its Appl. (ISITA), Auckland, New Zealand, Dec. 2008.