Algebraic space-time codes based on division algebras with a unitary involution

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May 2014, 8(2): 167-189. doi: 10.3934/amc.2014.8.167

Algebraic space-time codes based on division algebras with a unitary involution

Grégory Berhuy ^1,

1.	Université Joseph Fourier, Institut Fourier, 100 rue des maths, BP 74, F-38402 Saint Martin d'Hères Cedex, France

Received February 2013 Published May 2014

Abstract
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In this paper, we focus on the design of unitary space-time codes achieving full diversity using division algebras, and on the systematic computation of their minimum determinant. We also give examples of such codes with high minimum determinant. Division algebras allow to obtain higher rates than known constructions based on finite groups.

Keywords: space-time codes., algebras with involution, unitary involutions, Skew fields, algebraic coding.

Mathematics Subject Classification: Primary: 12E15; Secondary: 11T71,16W1.

Citation: Grégory Berhuy. Algebraic space-time codes based on division algebras with a unitary involution. Advances in Mathematics of Communications, 2014, 8 (2) : 167-189. doi: 10.3934/amc.2014.8.167

References:

[1]	G. Berhuy and F. Oggier, Space-time codes from crossed product algebras of degree 4, in Proceedings of Applied algebra, algebraic algorithms and error-correcting codes, 2007, 90-99. doi: 10.1007/978-3-540-77224-8_13.
[2]	G. Berhuy and F. Oggier, An Introduction to Central Simple Simple Algebras and their Applications to Wireless Communications, AMS., Providence, 2013.
[3]	G. Berhuy and R. Slessor, Optimality of codes based on crossed product algebras,, preprint., ().
[4]	B. Hochwald and W. Sweldens, Differential unitary space time modulation, IEEE Trans. Commun., 48 (2000), 2041-2052.
[5]	B. Hughes, Differential space-time modulation, IEEE Trans. Inform. Theory, 46 (2000), 2567-2078.
[6]	M. A. Knus, A. Merkurjev, M. Rost and J.-P. Tignol, The Book of Involutions, AMS, 1998.
[7]	F. Oggier, Cyclic algebras for noncoherent differential space-time coding, IEEE Trans. Inform. Theory, 53 (2007), 3053-3065. doi: 10.1109/TIT.2007.903152.
[8]	F. Oggier, A survey of algebraic unitary codes, in International Workshop on Coding and Cryptology, 2009, 171-187. doi: 10.1007/978-3-642-01877-0_15.
[9]	F. Oggier, J.-C. Belfiore and E. Viterbo, Cyclic Division Algebras: A Tool for Space-Time Coding, Now Publishers Inc., Hanover, USA, 2007.
[10]	F. Oggier and L. Lequeu, Families of unitary matrices achieving full diversity, in International Symposium on Information Theory, 2005, 1173-1177.
[11]	S. Pumpluen and T. Unger, Space-time block codes from nonassociative division algebras, Adv. Math. Commun., 5 (2011), 449-471. doi: 10.3934/amc.2011.5.449.
[12]	B. A. Sethuraman, Division algebras and wireless communication, Notices AMS, 57 (2010), 1432-1439.
[13]	B. A. Sethuraman, B. S. Rajan and V. Shashidhar, Full-diversity, high-rate space-time block codes from division algebras, IEEE Trans. Inform. Theory, 49 (2003), 2596-2616. doi: 10.1109/TIT.2003.817831.
[14]	A. Shokrollahi, B. Hassibi, B. M. Hochwald and W. Sweldens, Representation theory for high-rate multiple-antenna code design, IEEE Trans. Inform. Theory, 47 (2001), 2335-2367. doi: 10.1109/18.945251.
[15]	R. Slessor, Performance of Codes Based on Crossed Product Algebras, Ph.D thesis, School of Mathematics, University of Southampton, 2011.

show all references

References:

[1]	G. Berhuy and F. Oggier, Space-time codes from crossed product algebras of degree 4, in Proceedings of Applied algebra, algebraic algorithms and error-correcting codes, 2007, 90-99. doi: 10.1007/978-3-540-77224-8_13.
[2]	G. Berhuy and F. Oggier, An Introduction to Central Simple Simple Algebras and their Applications to Wireless Communications, AMS., Providence, 2013.
[3]	G. Berhuy and R. Slessor, Optimality of codes based on crossed product algebras,, preprint., ().
[4]	B. Hochwald and W. Sweldens, Differential unitary space time modulation, IEEE Trans. Commun., 48 (2000), 2041-2052.
[5]	B. Hughes, Differential space-time modulation, IEEE Trans. Inform. Theory, 46 (2000), 2567-2078.
[6]	M. A. Knus, A. Merkurjev, M. Rost and J.-P. Tignol, The Book of Involutions, AMS, 1998.
[7]	F. Oggier, Cyclic algebras for noncoherent differential space-time coding, IEEE Trans. Inform. Theory, 53 (2007), 3053-3065. doi: 10.1109/TIT.2007.903152.
[8]	F. Oggier, A survey of algebraic unitary codes, in International Workshop on Coding and Cryptology, 2009, 171-187. doi: 10.1007/978-3-642-01877-0_15.
[9]	F. Oggier, J.-C. Belfiore and E. Viterbo, Cyclic Division Algebras: A Tool for Space-Time Coding, Now Publishers Inc., Hanover, USA, 2007.
[10]	F. Oggier and L. Lequeu, Families of unitary matrices achieving full diversity, in International Symposium on Information Theory, 2005, 1173-1177.
[11]	S. Pumpluen and T. Unger, Space-time block codes from nonassociative division algebras, Adv. Math. Commun., 5 (2011), 449-471. doi: 10.3934/amc.2011.5.449.
[12]	B. A. Sethuraman, Division algebras and wireless communication, Notices AMS, 57 (2010), 1432-1439.
[13]	B. A. Sethuraman, B. S. Rajan and V. Shashidhar, Full-diversity, high-rate space-time block codes from division algebras, IEEE Trans. Inform. Theory, 49 (2003), 2596-2616. doi: 10.1109/TIT.2003.817831.
[14]	A. Shokrollahi, B. Hassibi, B. M. Hochwald and W. Sweldens, Representation theory for high-rate multiple-antenna code design, IEEE Trans. Inform. Theory, 47 (2001), 2335-2367. doi: 10.1109/18.945251.
[15]	R. Slessor, Performance of Codes Based on Crossed Product Algebras, Ph.D thesis, School of Mathematics, University of Southampton, 2011.

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