Characterization of $2\times 2$ full diversity space-time codes and inequivalent full rank spaces

Document Type : Research Paper


1 Department of Pure Mathematics‎, ‎Faculty of Mathematics and Computer‎, ‎Shahid Bahonar University of Kerman‎, ‎Kerman‎, ‎Iran.

2 Department of Pure Mathematics‎, ‎Faculty of Mathematics and Computer‎, ‎Shahid Bahonar University of Kerman‎, ‎Kerman‎, Iran

3 Young Researchers Society‎, ‎Shahid Bahonar University of Kerman‎, ‎Kerman‎, Iran.


‎In wireless communication systems‎, ‎space-time codes are applied to encode data when multiple antennas are used in the receiver and transmitter‎. ‎The concept of diversity is very crucial in designing space-time codes‎. ‎In this paper‎, ‎using the equivalent definition of full diversity space-time codes‎, ‎we first characterize all real and complex $2\times 2$ rate one linear dispersion space-time block codes that are full diversity‎. ‎This characterization is used to construct full diversity codes which are not derived from Alamouti scheme‎. ‎Then‎, ‎we apply our results to characterize all real subspaces of $M_{2}(\mathbb{C})$ and $M_{2}(\mathbb{R})$ whose nonzero elements are invertible‎. ‎Finally‎, ‎for any natural number $n>1$‎, ‎we construct infinitely many inequivalent subspaces of $M_{n}(\mathbb{C})$ whose nonzero elements are invertible.


Main Subjects

J.F. Adams, P.D. Lax and R.S. Phillips, On matrices whose real linear combinations are nonsingular, Proc. Amer. Math. Soc., 16 (1965), no. 2, 318--322.
S.M. Alamouti, A simple transmit diversity technique for wireless communications, IEEE J. Select. Areas Commun. 16 (1998), no. 8, 1451--1458.
M.D. Atkinson and R. Westwick, Spaces of linear transformations of equal rank, Linear Multilinear Algebra 13 (1983) 231--239.
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. Inform. Theory 51 (2005) 1432--1436.
A. Causin and G.P. Pirola, A note on spaces of symmetric matrices, Linear Algebra Appl. 426 (2007) 533--539.
M.O. Damen, A. Tewfik and J.C. Belfiore, A construction of a space time code based on the theory of numbers, IEEE Trans. Inform. Theory 48 (2002), no. 3, 753--760.
G.J. Foschini and M. Gans, On the limits of wireless communication in a fading environment when using multiple antennas, Wireless Pers. Commun. 6 (1998) 311--335.
R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge Univ. Press, 2nd edition, New York, 2012.
H. Kan, X. Liu and G. Han, On the criteria for designing complex orthogonal space-time block codes, Sci. China Inf. Sci. 59 (2016), 11 pages.
H. Khodaiemehr and D. Kiani, High-rate space-time block codes from twisted Laurent series rings, Adv. Math. Commun. 9 (2015), no. 9, 255--275.
X.B. Liang, Orthogonal designs with maximal rates, IEEE Trans. Inform. Theory 49 (2003), no. 10, 2468--2503.
M. Lucia and E. Mezzetti, Vector spaces of skew-symmetric matrices of constant rank, Linear Algebra Appl. 434 (2011) 2383--2403.
R. Meshulam, On k-spaces of real matrices, Linear Multilinear Algebra 26 (1990) 39--41.
A. Prestel and D. Delzell, Positive Polynomials, From Hilbert's 17th Problem to Real Algebra, Springer Monogr. Math. Springer-Verlag, Berlin, 2001.
J. Seberry, S. A. Spence and T. A. Wysocki, A construction technique for generalize complex orthogonal designs and applications to wireless communications, Linear Algebra Appl. 405 (2005) 163--176.
B. A. Sethuraman, B. Sundar Rajan and V. Shashidhar, Full-diversity, highrate space-time block codes from division algebras, IEEE Trans. Inform. Theory 49 (2003), no. 10, 2596--2616.
S. Sezginer, H. Sari and E. Biglieri, On high-rate full-diversity 2×2 space-time codes with low-complexity optimum detection, IEEE Trans. Inform. Theory 57 (2009), no. 5, 1532--1541.
V. Tarokh, N. Seshadri and A. Calderbank, Space-time codes for high data rate wireless communications: Performance criterion and code construction, IEEE Trans. Inform. Theory 44 (1998) 744--765.
S. Tofigh, H. Momenaee Kermani and A. Morsali, A new design criterion for linear receiver STBCs based on full-rank spaces, IEEE Commun. Lett. 19 (2014), no. 2, 1089-- 7798.
R. Westwick, Spaces of matrices of fixed rank, Linear Multilinear Algebra 20 (1987) 171--174.
R. Westwick, Examples of constant rank spaces, Linear Multilinear Algebra 28 (1990) 155--174.