American Institute of Mathematical Sciences

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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

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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

M. O. Damen, A. Chkeif and J. C. Belfiore, Lattice code decoder for space-time codes, IEEE Commun. Lett., 4 (2000), 161-163. Google Scholar

[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.  Google Scholar

[6]

P. K. Draxl, Skew Fields, Cambridge Univ. Press, Cambridge, 1983. doi: 10.1017/CBO9780511661907.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

T. Hanke, An explicit example of a noncrossed product division algebra, Math. Nachr., 271 (2004), 51-68. doi: 10.1002/mana.200310181.  Google Scholar

[12]

T. Hanke, A twisted Laurent series ring that is a noncrossed product, Israel J. Math., 150 (2005), 199-204. doi: 10.1007/BF02762379.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[15]

I. N. Herstein, Non-Commutative Rings, Math. Assoc. Amer., Washington, 1968. Google Scholar

[16]

B. M. Hochwald and S. T. Brink, Achieving near-capacity on a multiple-antenna channel, IEEE Trans. Commun., 51 (2003), 389-399. Google Scholar

[17]

T. W. Hungerford, Algebra, 3 edition, Springer-Verlag, Washington, 1980. Google Scholar

[18]

N. Jacobson, Basic Algebra I, 2nd edition, Freeman, New York, 1985.  Google Scholar

[19]

T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0406-7.  Google Scholar

[20]

P. J. McCarthy, Algebraic Extensions of Filelds,, Dover Publications Inc., ().   Google Scholar

[21]

J. Neukirch, Algebraische Zahlentheorie, Springer-Verlag, Berlin, 1992. Google Scholar

[22]

R. S. Pierce, Associative Algebras, Springer-Verlag, Berlin, 1982.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[32]

E. Telatar, Capacity of multi-antenna Gaussian channels, Europ. Trans. Telecommun., 10 (1999), 585-595. Google Scholar

[33]

J. P. Tignol, Generalized crossed products, in Séminaire Mathématique (nouvelle série), UniversitéCatholique de Louvain, Belgium, 1987. Google Scholar

[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.  Google Scholar

show all references

References:
[1]

P. M. Cohn, Introduction to Ring Theory, Springer-Verlag, London, 2000. doi: 10.1007/978-1-4471-0475-9.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[4]

M. O. Damen, A. Chkeif and J. C. Belfiore, Lattice code decoder for space-time codes, IEEE Commun. Lett., 4 (2000), 161-163. Google Scholar

[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.  Google Scholar

[6]

P. K. Draxl, Skew Fields, Cambridge Univ. Press, Cambridge, 1983. doi: 10.1017/CBO9780511661907.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

T. Hanke, An explicit example of a noncrossed product division algebra, Math. Nachr., 271 (2004), 51-68. doi: 10.1002/mana.200310181.  Google Scholar

[12]

T. Hanke, A twisted Laurent series ring that is a noncrossed product, Israel J. Math., 150 (2005), 199-204. doi: 10.1007/BF02762379.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[15]

I. N. Herstein, Non-Commutative Rings, Math. Assoc. Amer., Washington, 1968. Google Scholar

[16]

B. M. Hochwald and S. T. Brink, Achieving near-capacity on a multiple-antenna channel, IEEE Trans. Commun., 51 (2003), 389-399. Google Scholar

[17]

T. W. Hungerford, Algebra, 3 edition, Springer-Verlag, Washington, 1980. Google Scholar

[18]

N. Jacobson, Basic Algebra I, 2nd edition, Freeman, New York, 1985.  Google Scholar

[19]

T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4684-0406-7.  Google Scholar

[20]

P. J. McCarthy, Algebraic Extensions of Filelds,, Dover Publications Inc., ().   Google Scholar

[21]

J. Neukirch, Algebraische Zahlentheorie, Springer-Verlag, Berlin, 1992. Google Scholar

[22]

R. S. Pierce, Associative Algebras, Springer-Verlag, Berlin, 1982.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[32]

E. Telatar, Capacity of multi-antenna Gaussian channels, Europ. Trans. Telecommun., 10 (1999), 585-595. Google Scholar

[33]

J. P. Tignol, Generalized crossed products, in Séminaire Mathématique (nouvelle série), UniversitéCatholique de Louvain, Belgium, 1987. Google Scholar

[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.  Google Scholar

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