High-rate space-time block codes from twisted Laurent series rings

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August 2015, 9(3): 255-275. doi: 10.3934/amc.2015.9.255

High-rate space-time block codes from twisted Laurent series rings

Hassan Khodaiemehr ^1, and Dariush Kiani ^1,

1.	Department of Mathematics and Computer Science, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Ave., Tehran 15914, Iran, Iran

Received December 2011 Revised February 2015 Published July 2015

Abstract
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We construct full-diversity, arbitrary rate STBCs for specific number of transmit antennas over an apriori specified signal set using twisted Laurent series rings. Constructing full-diversity space-time block codes from algebraic constructions like division algebras has been done by Shashidhar et al. Constructing STBCs from crossed product algebras arises this question in mind that besides these constructions, which one of the well-known division algebras are appropriate for constructing space-time block codes. This paper deals with twisted Laurent series rings and their subrings twisted function fields, to construct STBCs. First, we introduce twisted Laurent series rings over field extensions of $\mathbb{Q}$. Then, we generalize this construction to the case that coefficients come from a division algebra. Finally, we use an algorithm to construct twisted function fields, which are noncrossed product division algebras, and we propose a method for constructing STBC from them.

Keywords: Space-time block codes(STBC), iterated twisted Laurent series, noncrossed product algebras., twisted Laurent series, crossed product algebras.

Mathematics Subject Classification: 11T71, 68P30, 17A3.

Citation: Hassan Khodaiemehr, Dariush Kiani. High-rate space-time block codes from twisted Laurent series rings. Advances in Mathematics of Communications, 2015, 9 (3) : 255-275. doi: 10.3934/amc.2015.9.255

References:

[1]	P. M. Cohn, Introduction to Ring Theory, Springer-Verlag, London, 2000. doi: 10.1007/978-1-4471-0475-9.
[2]	M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner and K. Wildanger, KANT V4, J. Symbolic Comp., 24 (1997), 267-283. doi: 10.1006/jsco.1996.0126.
[3]	M. O. Damen, K. Abed-Merriam and J. C. Belfiore, Generalized sphere decoder for asymmetrical space-time communication architecture, IEEE Electron. Lett., 36 (2000), 16-20.
[4]	M. O. Damen, A. Chkeif and J. C. Belfiore, Lattice code decoder for space-time codes, IEEE Commun. Lett., 4 (2000), 161-163.
[5]	M. O. Damen, A. Tewfik and J. C. Belfiore, A construction of a space-time code based on number theory, IEEE Trans. Inf. Theory, 48 (2002), 753-760. doi: 10.1109/18.986032.
[6]	P. K. Draxl, Skew Fields, Cambridge Univ. Press, Cambridge, 1983. doi: 10.1017/CBO9780511661907.
[7]	U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comput., 44 (1985), 463-471. doi: 10.2307/2007966.
[8]	G. J. Foschini, Layered space-time architecture for wireless communications in a fading environment when using multi-element antennas, Bell Labs. Tech. J., 1 (1996), 41-59.
[9]	G. J. Foschini and M. Gans, On the limits of wireless communication in a fading environment when using multiple antennas, Wireless Personal Commun., 6 (1998), 311-335.
[10]	J. C. Guey, M. P. Fitz, M. R. Bell and W. Y. Kuo, Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels, IEEE Trans. Commun., 47 (1999), 527-537.
[11]	T. Hanke, An explicit example of a noncrossed product division algebra, Math. Nachr., 271 (2004), 51-68. doi: 10.1002/mana.200310181.
[12]	T. Hanke, A twisted Laurent series ring that is a noncrossed product, Israel J. Math., 150 (2005), 199-204. doi: 10.1007/BF02762379.
[13]	B. Hassibi and B. Hochwald, High-rate codes that are linear in space and time, IEEE Trans. Inf. Theory, 48 (2002), 1804-1824. doi: 10.1109/TIT.2002.1013127.
[14]	B. Hassibi and H. Vikalo, On the expected complexity of sphere decoding, in 35th Asilomar Conf. Sign. Syst. Comp., Pacific Grove, 2001, 1051-1055.
[15]	I. N. Herstein, Non-Commutative Rings, Math. Assoc. Amer., Washington, 1968.
[16]	B. M. Hochwald and S. T. Brink, Achieving near-capacity on a multiple-antenna channel, IEEE Trans. Commun., 51 (2003), 389-399.
[17]	T. W. Hungerford, Algebra, 3 edition, Springer-Verlag, Washington, 1980.
[18]	N. Jacobson, Basic Algebra I, 2nd edition, Freeman, New York, 1985.
[19]	T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0406-7.
[20]	P. J. McCarthy, Algebraic Extensions of Filelds,, Dover Publications Inc., ().
[21]	J. Neukirch, Algebraische Zahlentheorie, Springer-Verlag, Berlin, 1992.
[22]	R. S. Pierce, Associative Algebras, Springer-Verlag, Berlin, 1982.
[23]	B. A. Sethuraman and B. S. Rajan, An algebraic description of orthogonal designs and the uniqueness of the Alamouti code, in Proc. IEEE GLOBECOM (2002), Taipai, 2002, 1088-1092.
[24]	B. A. Sethuraman and B. S. Rajan, Optimal STBC over PSK signal sets from cyclotomic field extensions, in Proc. IEEE Int. Conf. Commun. (ICC 2002), New York, 2002, 1783-1787.
[25]	B. A. Sethuraman and B. S. Rajan, STBC from field extensions of the rational field, in Proc. IEEE Int. Symp. Inf. Theory (ISIT 2002), Lausanne, 2002, p. 274.
[26]	B. A. Sethuraman, B. S. Rajan and V. Shashidhar, Full-diversity, high-rate space-time block codes from division algebras, IEEE Trans. Inf. Theory, 49 (2003), 2596-2616. doi: 10.1109/TIT.2003.817831.
[27]	V. Shashidhar, High-Rate and Information-Lossless Space-Time Block Codes from Crossed-Product Algebras, Ph.D thesis, Indian Institute of Science, Bangalore, 2004.
[28]	V. Shashidhar, B. S. Rajan and B. A. Sethuraman, STBCs using capacity achieving designs from cyclic division Algebras, in Proc. IEEE GLOBECOM (2003), San Francisco, 2003, 1957-1962.
[29]	V. Shashidhar, B. S. Rajan and B. A. Sethuraman, Information lossless STBCs from crossed-product algebras, IEEE Trans. Inf. Theory, 52 (2006), 3913-3935. doi: 10.1109/TIT.2006.880049.
[30]	V. Shashidhar, K. Subrahmanyam, R. Chandrasekharan, B. S. Rajan and B. A. Sethuraman, High-rate, full-diversity STBCs from field extensions, in Proc. IEEE Int. Symp. Inf. Theory (ISIT 2003), Yokohama, 2003, p. 126.
[31]	V. Tarokh, N. Seshadri and A. R. Calderbank, Space-time codes for high data rate wireless communication: Performance criterion and code construction, IEEE Trans. Inf. Theory, 44 (1998), 744-765. doi: 10.1109/18.661517.
[32]	E. Telatar, Capacity of multi-antenna Gaussian channels, Europ. Trans. Telecommun., 10 (1999), 585-595.
[33]	J. P. Tignol, Generalized crossed products, in Séminaire Mathématique (nouvelle série), UniversitéCatholique de Louvain, Belgium, 1987.
[34]	E. Viterbo and J. Boutros, A universal lattice code decoder for fading channel, IEEE Trans. Inf. Theory, 45 (1999), 1639-1642. doi: 10.1109/18.771234.

show all references

References:

[1]	P. M. Cohn, Introduction to Ring Theory, Springer-Verlag, London, 2000. doi: 10.1007/978-1-4471-0475-9.
[2]	M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner and K. Wildanger, KANT V4, J. Symbolic Comp., 24 (1997), 267-283. doi: 10.1006/jsco.1996.0126.
[3]	M. O. Damen, K. Abed-Merriam and J. C. Belfiore, Generalized sphere decoder for asymmetrical space-time communication architecture, IEEE Electron. Lett., 36 (2000), 16-20.
[4]	M. O. Damen, A. Chkeif and J. C. Belfiore, Lattice code decoder for space-time codes, IEEE Commun. Lett., 4 (2000), 161-163.
[5]	M. O. Damen, A. Tewfik and J. C. Belfiore, A construction of a space-time code based on number theory, IEEE Trans. Inf. Theory, 48 (2002), 753-760. doi: 10.1109/18.986032.
[6]	P. K. Draxl, Skew Fields, Cambridge Univ. Press, Cambridge, 1983. doi: 10.1017/CBO9780511661907.
[7]	U. Fincke and M. Pohst, Improved methods for calculating vectors of short length in a lattice, including a complexity analysis, Math. Comput., 44 (1985), 463-471. doi: 10.2307/2007966.
[8]	G. J. Foschini, Layered space-time architecture for wireless communications in a fading environment when using multi-element antennas, Bell Labs. Tech. J., 1 (1996), 41-59.
[9]	G. J. Foschini and M. Gans, On the limits of wireless communication in a fading environment when using multiple antennas, Wireless Personal Commun., 6 (1998), 311-335.
[10]	J. C. Guey, M. P. Fitz, M. R. Bell and W. Y. Kuo, Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels, IEEE Trans. Commun., 47 (1999), 527-537.
[11]	T. Hanke, An explicit example of a noncrossed product division algebra, Math. Nachr., 271 (2004), 51-68. doi: 10.1002/mana.200310181.
[12]	T. Hanke, A twisted Laurent series ring that is a noncrossed product, Israel J. Math., 150 (2005), 199-204. doi: 10.1007/BF02762379.
[13]	B. Hassibi and B. Hochwald, High-rate codes that are linear in space and time, IEEE Trans. Inf. Theory, 48 (2002), 1804-1824. doi: 10.1109/TIT.2002.1013127.
[14]	B. Hassibi and H. Vikalo, On the expected complexity of sphere decoding, in 35th Asilomar Conf. Sign. Syst. Comp., Pacific Grove, 2001, 1051-1055.
[15]	I. N. Herstein, Non-Commutative Rings, Math. Assoc. Amer., Washington, 1968.
[16]	B. M. Hochwald and S. T. Brink, Achieving near-capacity on a multiple-antenna channel, IEEE Trans. Commun., 51 (2003), 389-399.
[17]	T. W. Hungerford, Algebra, 3 edition, Springer-Verlag, Washington, 1980.
[18]	N. Jacobson, Basic Algebra I, 2nd edition, Freeman, New York, 1985.
[19]	T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0406-7.
[20]	P. J. McCarthy, Algebraic Extensions of Filelds,, Dover Publications Inc., ().
[21]	J. Neukirch, Algebraische Zahlentheorie, Springer-Verlag, Berlin, 1992.
[22]	R. S. Pierce, Associative Algebras, Springer-Verlag, Berlin, 1982.
[23]	B. A. Sethuraman and B. S. Rajan, An algebraic description of orthogonal designs and the uniqueness of the Alamouti code, in Proc. IEEE GLOBECOM (2002), Taipai, 2002, 1088-1092.
[24]	B. A. Sethuraman and B. S. Rajan, Optimal STBC over PSK signal sets from cyclotomic field extensions, in Proc. IEEE Int. Conf. Commun. (ICC 2002), New York, 2002, 1783-1787.
[25]	B. A. Sethuraman and B. S. Rajan, STBC from field extensions of the rational field, in Proc. IEEE Int. Symp. Inf. Theory (ISIT 2002), Lausanne, 2002, p. 274.
[26]	B. A. Sethuraman, B. S. Rajan and V. Shashidhar, Full-diversity, high-rate space-time block codes from division algebras, IEEE Trans. Inf. Theory, 49 (2003), 2596-2616. doi: 10.1109/TIT.2003.817831.
[27]	V. Shashidhar, High-Rate and Information-Lossless Space-Time Block Codes from Crossed-Product Algebras, Ph.D thesis, Indian Institute of Science, Bangalore, 2004.
[28]	V. Shashidhar, B. S. Rajan and B. A. Sethuraman, STBCs using capacity achieving designs from cyclic division Algebras, in Proc. IEEE GLOBECOM (2003), San Francisco, 2003, 1957-1962.
[29]	V. Shashidhar, B. S. Rajan and B. A. Sethuraman, Information lossless STBCs from crossed-product algebras, IEEE Trans. Inf. Theory, 52 (2006), 3913-3935. doi: 10.1109/TIT.2006.880049.
[30]	V. Shashidhar, K. Subrahmanyam, R. Chandrasekharan, B. S. Rajan and B. A. Sethuraman, High-rate, full-diversity STBCs from field extensions, in Proc. IEEE Int. Symp. Inf. Theory (ISIT 2003), Yokohama, 2003, p. 126.
[31]	V. Tarokh, N. Seshadri and A. R. Calderbank, Space-time codes for high data rate wireless communication: Performance criterion and code construction, IEEE Trans. Inf. Theory, 44 (1998), 744-765. doi: 10.1109/18.661517.
[32]	E. Telatar, Capacity of multi-antenna Gaussian channels, Europ. Trans. Telecommun., 10 (1999), 585-595.
[33]	J. P. Tignol, Generalized crossed products, in Séminaire Mathématique (nouvelle série), UniversitéCatholique de Louvain, Belgium, 1987.
[34]	E. Viterbo and J. Boutros, A universal lattice code decoder for fading channel, IEEE Trans. Inf. Theory, 45 (1999), 1639-1642. doi: 10.1109/18.771234.

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